Hdn nffa vdi diTdng thing (d) thoa tinh chat cac giao di<^m ciia cac dufdng thing trong A^QOS ci cung mot nufa mat phing cd bd la (d) (luc nao trong Ajoojta ctlng chpn difdc mot d[r]
(1)TS. N G U Y E N CAM (Chu bien)
^ ThS. N G U Y E N V A N PHUCfC
REN LUYEN TOAN NANG CAO
DAI SO
* Phan loai cac dang toan nang cao dinh cho hoc sinh gi6i * Tuyen chon cac de thi tuyen sinh Idp 10 vao cac triTdng chuyen
THLT VIENTIfvHBINH THUAN
(2)LdlNOIDAU
Cung vdi viec bien soan bo sach Ren luy^n todn can ban LOP 9, chiing toi gidi tliieu cuon Ren luyen todn ndng cao DAI SO nham giup cac em hoc sinh nang cao trinh de tham gia cac ki thi tuyen vao Idp 10 thuQC cac trirdng chuyen, Idp chuyen
Cuon sach bao gom chifcfng : ChUdng : Dang thiJc ChiTdng : Bat dang thiJc Chufdng : So hoc
ChiTdng : Gia tri Idn nhat va gia tri nho nhat cua ham so ChiTdng : Phifdng trinh
ChUdng : He phifdng trinh ChiTdng : Do thi ham so
ChiTcfng : Mot so bai toan ve td hdp ; dpng tiJ ChiTdng : Mot so de thi vao Idp 10
Vdi muc dich h5 trd dac life viec tuT hoc ciia minh, cuon sach diTc^c trinh bay theo bo cue nhiT sau : md dau mSi chufdng la phan torn t^t cac kien thilc can ye'u ma hoc sinh phai nam vffng; tiep theo la mot he tho'ng b^i tap CO hiTdng din each giai diTdc sap xep theo thiJ tif kho dan
De viec hoc cd ket qua to't, sau doc va tim hieu that ky cac bai tap CO hiTdng din hoc sinh nen co gang tif minh giai cac bai tap ay mot cdch doc lap cho den thuan thuc Vdi each lam ay, hy vong cac em se tif minh kham pha nhieu each giai khac mot each thu vi
Raft mong nhan diJdc cac y kien phe binh cua quy doc gia de cuon sich cang diTdc hoan thien va phuc vu ban doc tot hdn
TM.nhdm tdc gid TS Nguyen Cam
(3)f
CHl/OfNGI: DANG THlfC
I HANG DANG THlfC
1) (a + bf =a'+2ab + b\
2) {a-bf=a' -2ab + b\
3) {a + bf =a' -h^a'b + ^ab^ +b\
4) {a-bf=a' -3a'b +3ah^-b\
5) a' +b' ={a + b){a'-ab + b')
6) a'~b'={a b)(a^+ab + b^)
7) a"-b"={a b\a"-' +a"-'b + a"-\b' + + ab"-' +b"-') 8) {a + b + cf = +b^ + 2(ab + bc + ca)
9) (a + Z) + c)' = a'+b'+c'+ 3(a + b)(b + c){c + a)
n. CAN THlfC
1) ^A^ =\A\=< AneuA>0
1) ^A^ =\A\=<
-AneuA<
2) A+B±lylAB=(^±4B^{A>0,B>Q)
3) ^ + 5' ± V I = ( V I± ) ' ( ^ > ) 1 -IA-JB
-j-j-(A>0,B>0,A^B)
1 •SIA+JB
(4)1.1: Tinh cac bieu thitc :
2 - V + V3
3) C = ^ + '
4) D = - l ^ ' 3 - V + V
1) A = - i ^ + ' 2 + V3
2 - ^
2) B =
3) C =
2-S + V3 f2-V3)(2 + ^/3] (2 + ^3) (2 - ^/3)
+ ^ + ^ = + 7^ + 2- 7^ ^ 4 - -
1 ^ / 5+ V V - V 2V5
• + + •
^5-42+ V + V - - V 3+ I V 3- I
V 3- I V 3+ I - - =
1 _3 + V2 3->/2 ^ + V 2+ - N ^ _
3- V^ ^ + V 5^ ^ - 2^ 3^ - = 7' De 1.2
1) Riit gon bieu thiJc M = a ;t va a >
I - A/ ^ l + ^/ajv ^/« - vdi
2) Tinh gia tri cua M a = -
6
1) M =
-,i->/^ i+V^Jt (i-V^)(i+V^)'~vr~ 2Va V a- 1 -
( l - V a ) ( l + V a ) ' V a 1 + V«
- -3 2) Khi a = - T a c : M = — ^
3
1.3: Rut gon (Loai bo dau can va dau gia tri tuyet do'i) 1) A = ^Ja^ - a +
2) = Vx^ + Vx' - x + Giii
1) A = ^Ja^ -6a + =^(a-3y =\a-3\ a-3 neu a-3>0 f a- 3 new a> 3 - a n ^ M a- < 3- a neu a<3 2) = V7 + V x' - x + = V7 + ^ ( x - ) '
= | x | + | x - |
-x-x + 2neu x<0 {-2x + ne'ux<0 x-x + neu0<x<2 = l2 neu 0<x<0 x + x-2neu x>2 2x-2 neu x>0
De^ 1.4: T i i 1) ^ =
2) B = ih:
V - V
L ^/a+l)
+ V O 8- - V '147
(5)Gi§i
1) yl = V48-2V75 + Vr08-iVT47 = - 2 V i ^ + V ^ - - V T ^
7
2) B = 1 +
1 +
1
-= ( l + V ^ ) ( l - V ^ ) -= l - a
1.5: Tinh :
rV216 2^-yf6^
[ 3 V 8- 2 , 1
7216 V - V ' ~ V -
V ^ ( ^ / ^ - l )
6^6 V - V
•3 - ^ - ) yfe
2 > /
-2( v ^ - 1"
1 _ 2 > / - 2>y6 V6 V6 76 2^/6
2
1.6: Tinh:
5 = 713-7160 - + 4790
8
= 78-2715 - + 2715 = - + - + 2715+3
5) - 27^7^ + (73) - ^(7?) + 27^^^ + ( ^ ) '
= ^ ( ? - ^ f - ^ ( ? + ^ f
= 75->^|-|75 + 73| = 75 - 73 - - ^ ( v / > ) = -
5 = l - l - 753 + 4790 = 78 - l o" f -7 + 4790+
8 ) ' - ^ + (75 ) ' - -^(745 ) ' + 2745 78 + (78 ) ' = ^ ( > ^ - ) - ^ ( - ) =|78 - 75|-|745+78 = 78 - 75 - - ^ (v/ > ^ > )
= - - = -475
1.7: Cho bieu thitc:
M =
7^ •
1 + ^ 27a
7 a - l (a + l ) ( a - l ) 1) T i m dieu kien cua a de M CO nghia 2) Rut gon bieu thu-c M
3) V d i gia tri nguyen nao cua a thi M c6 gia tri nguyen ?
M = 1:
1:
-1 + ^
1 27a
\ yfa
(6)1: l + \/aj (a + l)(x/a-l) a>0
1) Bieu thu'c M c6 nghia <=> \ 4a^0 \a>0 <=> { 2) Vdi dieu kien a > va \a c6:
' ^ ( v / ^ - l j (l + V^)(-v/^-lj
a >
1: + VaJ (a + l)(Va-l) (a + l)(Va-l)
a + \ a +
3) Ta c6: M = -^-^ = - — B i e u thu'c M nhan gia tri nguyen a + a +
khi a >
a ;t <z> 2;(a + l)
a >
a 1 o
a + l = l;-l;2;-2
a >
a = 0;-2;l;-3 Vay a = thi bieu thu'c M nhan gia tri nguyen 1.8: Cho bieu thu'c:
M = 4^a'~a'b'
M =
b' vdi |a| > |^>| >
Rut gon bieu thu'c M Vdi dieu kien a\>\b\> Tacd:
10
a' + l a ^ l a ^ + a^-b^-a^+ la^a"-b"-a" + b" 4|a|Va^-6'
a^-[a'-b') b' Aa^a'-b' Aab'^a^-b' _ fl mu a>Q
'A\a\4^^ 4\a\b^yja^-b^ [-1 neu a<Q DC 1.9: Riit gon bieu thu'c:
3 (yfai) - b) (yfa - f + 2av'a + b ^ M=-^ ^- + ^ ^= P
a-b ay/a+by/b (vdia>0, b>Ova a;tb)
V d i a > , b>Ova a^^b, ta c6:
3(V^-Z>) (4a-4b^ +la4a+bylb
M = a-b aja+byjb • + •
3V^(V^-V^) ayf^-3ayfb+3by[^-by[b+2ayl^ + b ^ (V^-V*)(N/^ + V ^ ) ^ [ ^ + yfb)(a-yf^ + b)
3yfb ^ 3ay[a-3ay[b+3by/a (yf^ + yfb) (^/^ + ^ / ^ ) ( a - ^ / ^ + Z>)
3jb 3yfa(a-yfab +b)
(V^ + VA) (^ + y[b^(a-4ab+b^
34b , 3ra _3V^ + V ^ _ ( ^ + ^ ) _ y/a+yfb yfa+yjb yja + yfb yfa + yfb
(7)1.10: Rut gon bieu thrfc sau:
- yfa a\[a + a + -Ja
1) ^ =
- >/a ayfa + a + ^/a
1
'a>0
>/a + 0 yfa ^\
V d i dieu k i e n a > va a;t ta cd:
1 >/a+l
B i e u thitc A cd nghia o a>
A =
^/a(a + ^ / a + l j j
2) B = ^Jx^ +2x + \-ylx^ -2x + \
=^{x+\f-^{x-\y = \x+\\-\x-\\
- x - l - ( - x + l ) neu x < - l
x + l - ( - x + l ) neu - l < x < l = x + l - ( x - l ) neu x>\
- 2 neu' x < - l 2x neu' - l < x < l 2 wew' X >
D e 1.11: Cho M = Vx +
X^ -4x Xyjx + X + y[x
12
1) T i m dieu k i e n cua x de M cd nghia 2) Rut gon M
1) M cd nghia o \
Gidi X >
x ^ - ^ ^ yfx+l^O
x-Jx + x + ^fx ^
x >
^ / x ( x + ^ / x + l ) ^
x>
{•fx)'-wo
x> x>
x;tl
2) V d i d i ^ u k i e n : x > va x ^ Ta cd :
1 Vx + 1 > / x + l
X^-yfx xJx+X + yfx >/x(x + Vx + l)
1 ^[x + yf^ + l) ^ ^ ^
V^(V^-l)(x + V^ + l)' + \^ + l x-\
D e 1.12: Cho M = 4 x - \ - i ^ V x^ -
1) T i m dieu k i e n c\ia x de M cd nghia
2) T i n h M l 3) Rut gon M
1) M e d nghia
GiSi
x - >
x - l - ^ / x ^ > o
V T ^ - l ^ O
x >
( x - ) - > / x ^ + l ^
yfx^^l
(8)x>2
( V x - - l ) ' > <=> x>2 x^3 x - l ^ \
2) V d i d i ^ u kiSn x > va x 3, ta c6 : V X- - 2A / X ^
Vx - -
\ X -1 - 2Vy -
( x- ) - V ^ + l • - l- V > : -
= x - l- x ^ 3) V d i = ^ M = ±1
M a t khac v i : y]x-\-2y[x^ > nen :
• Ne'u : V x - - > <::> V x - > l o x - > l < : > x > thi M > ; d o d : M =
• Ne'u : V x - - l < < r > v ' x - < l < = > < x - < l < : : > < x < t h i M < ; d d : M = - l
1 (neu : X > 3)
- ( n^ M: < x < ) vonghia {neu: x = 3;x < 2) V a y M =
D e 1.13: Rut gon cac bieu thiJc:
x V x ^ + Gi§i
3- J ( V 5- )
= ^ / ^ / ? ^ V 5- ^ / + l = J V S - J V s ) -2V5.1 + 1^
= ^ V - ^ p ^ = ^ V - | V - l | = ^ V - ( V - l ) (V/ V5 >1)
y =VV5-V5+i=vr=i
2) =
+ x ^ + ( x ^ + x ' ' + ) - x ^ x ' + x' + 2 x ' + x' +
( x V 2) ' - ( x ^ ) ' ( / + 2 + x^ ) ( x V 2- x ^ ) x V x' + 2 " x V x +
= x ^ - x ^ +
1.14:
u-^ , ^ 1 V 2- ^
1) R , U g p n b e u t h c M = - ^ ^ ^ ^ ^ 2) T i m gia t r i nho nha't cua:
y = -1 - V x ^ + Vx + T - W x ^ Gi^i
1) M = - I3V2-2V3
V 2- V V3V2 + 2V3 V - V 3^ V 6( > / + >/2)
1
V 2- V ^ -
1
3-
( > ^ - V ) ( v i V > V )
2) Ta cu :
(9)= ylx-2-2y/x-2+l+yjx-2-6^fx^ +
Dieu kien: x>2
Ap dung \A\>A Da'u "=" xiy o ^ > Ta c6 y> •Jx-2 -1 + - V x - = •
V x ^ - >
+
Da'u xay o <^
3 - V x - >0
o l < x - < o < x < l l Vay Miny = 2,datdu'c{ckhi:3<x<ll
<=>\<sfx^ <3
Bi 1.15: Phan tich nhan IvC M = x'' +x' +1
M = x"'+x'+l
= x'° +x' +x'-x'-x'-x' + x' +x' +x'-x'
-X^ -X* +x^ +x^ +x^ -x^ -x^ -x + x^ +x + \ = x'(x'+x + l)-x'(x'+x + \)+x'(x'+x + l)
-x" (x^ + X +1)+ x'(x^ +x + l)-x(x^ +x + l)+l(x^ +x + l) = (x'+x + l X x ' - x U x ' - x V x ' - x + l )
1.16: Chitng minhr^ng ne'u: -ab-bc-ac = 0, thi: a = b = c
GiSi Tacd: -ab-bc-ac =
O l(a^ - lab + b') + ^{b'-2bc + c') + l(c^ - lac + a')
1 1 1
<>-(a-bf+-(b-cf+^c-ay=0
2 ^ 16
a-b =
b-c-O <;i>a = b = c c - a =
Vay, nSu a^ +b^ -ab-bc-ac = thi a = b = c )e 1.17: Cho ba so' a, b, c cho c ;t b, c ?t a + b va
=2iac + bc-ab) (1) ^-{a-cy _ a - c ChiJng minh rkng:i
b^+{b-cf b-c GiSi
TO (1) ta c6: + 2aA - 2ac - 2^)C = nen: = a^ + {c^ + lab ~ lac - 2bc) s = (a^ - 2ac + ) + 2Z)(a - c)
= {a-cf+lb{a-c) = {a-c){a-c + lb) Wngtuf: b'^ =b^+(c^+lab-lac-lbc^
= [b^ -lbc + c^) + la(b-c)
= (b-cy +laib-c) = {b-c)(b-c + la) a' + {a-cf _ (a - c){a-c + 2A)^+ (a - cf Do do:
b'+ib-cf ~ ib-c)(b - c + la) + {b- cf (a-c)(la-lc + lb) _a-c
~ (b-c)(2b-lc + la) b-c (vi a+b^c) D6 1.18: Cho a, b, c doi mot khac va thoa
a b c ^ + = b-c a - c a-b
(1)
(10)TO (l)ta c6:
Gi§i
a b c ab-b^ -ac + c^
b-c a-c a-b (a-c){a-b) a ab-b^ -ac + c^
(b-cf {a-c){a-b){b-c)
Lam Wdng W nhtf tren, ta se c6:
b _ bc-c^ -ba + a^
(c-af " (a-b){a~c)(b-c) c ca-a^ -bc + b^
{a-by ~ (a-b)(a-c)(b-c)
Cong ve theo ve (2), (3) va (4) ta diTcJc:
a b c • + r + r- =
(2) (3) (4)
(b-cf {c-af (c-bf
De 1.19: Cho ba so' a, b, c thoa a + b + c = abc
Chu-ng minh: a(A' - l)(c' -1) + b(a^ - l)(c' -1) + c(a' - \)(b^ -1) = 4abc
GiSi
Ta CO: a(b^ - l)(c' -1) + b(a^ - l)(c' -1) + c{a^ - \)(b^ -1)
= ia + b + c) + (ab^c^ + baV + ca^b^) - (ab^ + ba^ +ca^+c^a+be" + b'^c) = abc + {ab^c^ + ba^c^ + ca^b'^) - [ab{b + a)+ ac{a + c) + bc{b + c) = abc + (ab^c^ + ba^c^ + ca^b^) - [ab(abc -c)+ac{abc -b) + bc{abc - a)
(vi a + b + c = abc nen a+b-abc-c,b + c-abc-a,a + c = abc-b) = abc + (ab^c^ + ba^c" + cc^b^) - (a^Z»'c+a'c^i+ab'^c^)+l>abc = 4abc
Vay ddng thiJc da diTdc chiJag minh
1.20: Cho - = ^ = - = w (m>0). Chiang minh ring : a b c
^a'+b'+c'
18
Tacd: — = — = — = m cho ta: m
=—j a b c =—j a b a''+b^^c'
nen: m = ^a'+b'+c'
Bi 1.21: Choa,b,c, a',b',c' la cac so diTdng thoa — = — =
a' b' c' CMng minh: ^|aa' + ylbb* + yfcc' ^ yj(a + b + c)(a'+b'+c'j
GiSi Dat: w = — = — = — thitac6: a' b' c'
a + b + c (a + b + c)(a'+b'+c') ~ a'+b'+c'~ (a'+b'+cf
^ , aa' bb' cc' r- 4aa' JbT' yfcc'
Talaico: m = —- = —- = —-=> y^m^ ^- =
a'' b'^ c ' ' a' b' c'
[— yfaa'+ ^/bT'+ yfcc'
=>ylm
a'+b'+c'
TOCl) va (2)tasuy ra: {a + b + c)(a'+b'+c') yfaa' + \fbb' + \[cc
(a'+b'+cf
nen {a + b+ c){a'+b'+c') = (4^'+4bb'+ 4^'^ a'+b'+c'
2
hay 4aa'+ y]bb'+y[cc'= ^{a + b + c)ia'+b'+c') (dpcm)
(1)
(2)
De 1.22: Cho — = —= ^ vd — + ^ + — =
X y z a^ b^ c^'
Churng mmh: — + — + = — — ^
a" b^ c" x^+y^+z"
(11)Dat / = — = —= — thico x = —;)> = —;z = —
X y z t t t
a^t^ b^t^ c^f
2 rn'
TO / = — = - = — thi cd: =
X y z
TO (1) va (2) ta suy r^ng:
a'
x' / z'
(1)
x^ ^y^ -v z^ (2)
p^ + p^ (dpcm)
1.23: Cho ba so' a, b, c thoa - a'+b'+c'^l (1)
[a^+b'+c^=\) my iiDhtSng so S ^a + b^+c\
Gidi
TO(2)tac6 a' <l,b^ < l , c ' <1 nen: |a| < 1, \b\ 1, |c| <1
=>a' <a\b' <b\c' <c'
m a ( l ) v a (2) cho ta: a' +b' +c' +b^ = nen phai xay b'=b'
\6i =a^ => a^(a-l)-0=> a = hay a-l=>a^=a
vadodd S = a + b^+c^ =a^ +b^ = \ VayS = l
20
CHLfOfNGll: B A T DANG THtfC
§1 PHEP BIEN DOI TUdNG DLfdNG TINH CHAT CUA BAT DANG THLfC I PHEP B I E N DOI Tl/dNG DlJdNG
1) 2)
a>b <:i> a-b >Q a>b <^ a-b>0
Chu v; Vdi moi so' thrfc X, ta cd: >
II C A C TINH C H X T C d BAN CUA BAT DANG THLfC 1) a>b va b>c^a>c
2) a>boa + c>b + c 3) a>b + c<^a-c>b
a>b
4)
c>d
5) a>bO'
'a>b>0
6) c>d>0
•a + o b + d ac > be ni'u c >
ac<bc ne'u c<Q ac > bd
7) Neu:ab>0 thi:a>bo-<-
a b
Bi 2.1: Cho a > 0, b>0 CMng minh: a^ +b^ >a^b + ab^
Giii
Tacd: a' +b^ >a^b + ab^oa'-a^b + b'-abJ>Q
(12)<:>a\a-b)-b\a-b)>0<>(a^-b^)(a-b)>0
o(a + b)(a-b)(a-b)>0<:>ia + b)(a-by >0 (*)
V i a > , > nen a + b>0 va (a-bf >0 n^n (*) diing Vay +b^ >a^b + ab^
Bi 2.2: Cho a > 0, b > Chitng minh: + > ^/a + V ^
yjb Va
T a c d : -^^-^>4a + y[b<:>a4a + b4b >(y[a+y[b)^fab
yfb ^ ^ '
i^^J + ( V * ) ' - ( V ^ +V A) V o ^ >
[4^+y}b)[4^-y[^+Jh'Y[4^+4b)4ab>o •
o
o ( + ^/6 ) ( - + V * ^ ) > (N/^ + V ^ ) ( V « - V * ) ' > (diing) Vay: - ^ + - ^ > y/a + yfb
yjb yja
Bi : Chitng minh ba't dang thitc sau: +l>ab + a + b v d i
moi so'thifc a, b
GiSi
T a c d a ' +Z»' + > aA + a + A o 2a'+2Z>^ + > 2aA+ 2a + 2Z> <=>2a' + ' + - a - a - Z ) >
o ( a ' -2a^)) + ( a ' + l - a ) + (6' + - Z > ) > o ( a - ) ' + ( a - ) ' +{b-iy>0 (diing) Vay a ' + ' + l > a Z » + a +
f)i 2.4: Chitng minh cac ba't dang thtfc:
1) a " * > a^6 + aZ>^ vdi moi a, b 2) a ' + ' > - v d i a + >
GiSi
1) T a c d a*+b^ >a'b + ab' <^ + b^ - a'b - ab' >
<^a'(a-b)-b'(a-b)>0^{a-b)(a'-b^)>0
o (a - ^»)(a - Z))(a' + aA + A') ^ <^ (a - 6)' a ' + a + — + 36 2 A > < : > ( a - ) ' a + - + -36' > (dung)
Y&y: a'+b'>a'b + ab\
2) T a c d a ' + ' + 2a6 = (a + ) ' > l ( l ) va -lab = [a-bf >0 (2) Cong (1) va (2) ta cd a ' + ' > 1 a ' + ' > i
V a y a ' + 6' > i
Bi : Chu-ng minh bat dang thiJc:
a ' + ' + c ' > a + 6c + c a GJdi
Ta cd: a ' + ' + > a6 + c + ca O a ' + ' + c ' ^ 2a6 + 26c + 2ca
<:>(a' - a + ' ) + (6' - c + c ' ) + (c' - c a + a ' ) > O (a - ) ' + (6 - cf + (c - a ) ' > (diing)
(13)D6 2.6: Chu'ng m i n h rang vdi m o i a, b , c, d, e ta c6:
(a^ >a{b + c + d + e)
Gidi
T&cd: + b ^ > a { b + c + d + e) <^a^+b^+c'+d^ + e^-a(b + c + d + e)>0
<^a^ + b^ + c^+d'^ + e'^-ab-ac-ad-ae>0
t: + b^-ab + a • + c'^-ac + - + d^-ad + Ve —ae
4
>
^—b
2 +
a
c
2 / \1
2
+ + a e
2-2 '
> ( d u n g ) V a y : + + + c/^ + > a(A + c + J + e )
D d 2.7: Chtfng m i n h rang vdi m o i a, b, khac ta c6:
b' a'+b'<^ +
a ,2 •
a* b^
T a c d : a ' < — + — •
b' a'
O a^t' + a'b' <a'+b'<^a'- a'b' + b'- a'b' >
O a'(a' - fc') - b\a^ -b')>Oo(a'- b')ia' -b')>0 o(a' -b')ia' -b'){a' +a'b'+b')>0
<:>ia'-b')\a'+a^b^ + b')>0 (dilng)
V a y : a +b' <^ + ^ b a
6 ^
D d 2.8: C h o a.b>0. Chu'ng minh rang: (a + bfxy<{ax + by)(bx + ay)
24
T a c(5: (a + b)"" xy<(ax + by)(bx + aj)
<=> a^xy + 2aZ7jc>' + b^xy < abx^ + aby^ + c^xy + b^xy <:> abx^ + aby'' - 2aZ?x>' > <::> ab{x^ + -2^:^) >
<^ab{x-yf>Q
Ba't d^ng thi?c n a y diing vi ab > 0,(jc - y)^ > V a y : {a + bf xy <{ax-\- by)(bx + ay)
Bi 2.9: C h o a + > Chu'ng m i n h rang: a^ +b' <a' +b\
Gidi
TsiCd: a'+b'>a'+b^ oa'-a'+b'-b'>0
c>a\a-l) + b\b-l)-{a-\)-ib-l) + a + b-2>0 <^(a-l)(a'-l) + (b-\)(b'-l) + a + b-2>0
<:>(a-l)\a^ + a + l) + (b-l)\b' + b + l) + (a + b-2)>0
Bat d d n g thiJc tren d u n g vi: •
* (a-lf >0,(b-\)^ >0
a^+a + \
2 * a + ^ - > 0 (do:a + b>2)
V a y : + Z?^ < a' + Z?^ da du'dc chu'ng m i n h
D d 2.10: C h o ba so a, b, c thoa a' + Z?' + =
Chu'ng m i n h rang: abc + 20- + a + b + c + ab + ac + bc)>0
(14)Gidi
Tac6: a' = nen <\,b^ < l , c ' < ^ - < a < l , - l < ^ < l , - l < c < l
^ l + a > , l + Z 7> , l + c > => il + a)il + b)(l + c)>0
=^l + a + b + c + ab + bc + ca + abc>0 ' (1) Mat khac, ta lai cd: (1 + a + + c)^ >
^\ a^ +b^ + +2(ab + bc + ca) + 2(a + b + c)>0
=>2 + 2(ab + bc + ca) + 2(a + b + c)>0 (vi a^ + b^ =1) =^l + ab + bc + ca + a + b + c>0 (2)
Cong ve theo ve' (1) va (2) ta difdc:
abc + 2(l + a + b + c + ab + bc + ca)>0(dpcm)
De 2.11: Cho < a < l , < Z ? < l va < c < LChu-ng minh:
+ b ^ < l + a^b + b^c + c^a (1)
GiSi
Tacd: (l)<::>a^(l-Z?) + Z>^(l-c) + c ' ( l - a ) < l ma < a < 1,0 < < 1,0 < c < nen :
a\l-b)<ail-b) b\l-c)<b(l-c) c\l-a)<c(l-a)
Do de chiing minh (1) ta chi can chiJng minh bat ddng thrfc sau:
a(l-b) + b(\-c) + c(\-a)<\)
That vay, ta c6: - a > 0,1 - ^ > 0,1 - c >,a^c > nen (1 - a)(l - ^)(1 - c) + a^c >
=^l-{a + b + c) + ab + bc + ca>0 ^l>a + b + c-(ab + bc + ca) =^ a(\ + b(l-c) + c(l-a)<l
Do d6 (2) dufcJc chiJng minh Vay bat d^ng th-ic (1) diTcJc chu-ng minh
1
pi 2.12: Cho ac> va thoa : - + - = -a c (1)
, a + b 'c + b ^
Chiang mmh : + >
2a —b 2c —b
Giii
2ac
c + a
Tuf(l)tac6: - = ^b =
b ac a + c
do dd: - + •
2ac
a + c c + lac a + c 2a-b 2c-b 2a- 2ac
a + c
2c-2ac a + c a^+3ac c'^+3ac
2a' 2c'
( 3c' r 3a' , c a l + — + - \ — + -[ a J 2 [ c J 2 [a cj V i a c > O n e n - > va - >
Tacd: J - - J - c a r (1
> = > - + - - > ^ - + - > a c a c
>1.2 = Vay : ^ + ^ >
~2 2a-b 2c-b~ Bi 2.13: Cho a > , b >0, c>0thoa a + b = c Chitngminhr^ng:
I Giii
Ta cd: a > 0, b > va a + b = c nen < a < c v a O < b < c
a a , b
> - = vd —!=>
a b a b a" b' a + b c , , + i—>—;=- = - = (vi a + b = c) '4b 'Tc fc \a "ib 4~c fc
(15)(1) 2.14: Cho ba s6' a, b, c thoa: ab + bc + ca>0 (2)
aboO (3)
Chiang minh rang a > 0, b > va c>
Giii
Ta chiJng minh bang phan chitng: Gia su* c6 : a <
Tir (1) => + c > - a > nen b + c>
Tv[(2) ^a(b + c) + boo
=>bc>-a(b + c)>0 (vi a<0,b + c>0)
Do a < va b o nen suy abc < Dieu trai vdi gia- thi6'l la abc >
Trirdng hop b < hoac c < 0, ta ly luan trfdng tif Vay ta da chiJng minh r^ng: a > 0, b > 0, c>
Bi 2.15: Cho < a < l , < b < l , < c < l Chitng minh ba bat
dang thtJc sau cd it nha't mot bat dang thiJc la sai:
ail-b)>]-, b(l-c)>]-, c(\-a)>]- 4 4
Gi^i
Ta chiJng minh bang phan chiJng:
Gia su" ca ba bat ding thiJc deu dung, liic dd nhan ve'theo ve ta cd:
a(l-a)bil-b)c(\-c)>
^
(1) Mat khac ta cd:
V ^ - V T ^ ) ^ >0=>a + ( l - a ) - ^ a ( l - a ) > 28
V a ( l - a ) < i = ^ a ( l - a ) < i TuTdng tir ta cd :
b(l-b)<
c ( l - c ) <
]_
4 4
Nhan ve'theo ve ba ba't ding thtfc vuTa n6u thi:
a(l-a)b(l-b)cO.-c)<
Dieu mau thuan vdi (1)
Vay ta phai cd: Trong ba bat dang thiJc di bai cd it nhat mot bat dang thiJc sai
(16)§ BAT DANG THLfC COSI (Cauchy)
I BS't d a n g thufc C6si cho h a i so' k h o n g fim C h o a > , & > :
Ta c6: a + b > ^ Da'ii dang thiJc xay ra ^ a = b
I I Ba' t d a n g thuTc C6si cho n s6' k h o n g &m
Cho n so'khong am: a^,a^,ããã,ô ã T a c :
Dau dang thiJc xay <^ = = = a „
D I I : Cho a, b, c > Chitng minh:
2) (a + fe + c) Da'u dang thiJc xay k h i nao ?
1) Ut>2
b a a b c >
GiSi
1) A p dung bat dang thiJc Cosi cho hai so' khong a m - va - , ta c6: b a
b a
a b a b ^ ^
- - ^ - + ->2 (dpcm) \b a b a
Da'u d i n g thiJc xdy ra <^ — = -<F^a = b b a
2) A p dung B D T C6si cho ba so' khong am, ta c6:
• a + b + c>3y[abc (1)
i , U i > f i i
a b c \ b c
(2)
30
Nhan (1) va (2) ve theo ve', ta dtfcJc: (a + b + c)
a b c > L c~ V a b c .i.i.i
Da'u dang thiJc xay <^
a b c > (dpcm) a = b = c
1 1 1 a = = c a b c
D e 2.17: Cho a, b, c > Chitng minh: [• + -1 > c, I a) Da'u dang thiJc xay k h i nao ?
A p dung B D T Cosi cho hai so'khong am, ta c6: (1)
l + - > J l ^
b \
c V c
l + - ^ > i £
a \
(2)
(3)
Nha.i (1), (2), (3) v e t h e o ve'ta dUdc:
1 + ^ 1 + ^' [1 + - ] 1 + ^
c K "1 \b c a i b] c,
> (dpcm)
D f u d i n g thiJc xay <^
1 = ^
\ a^b = c
a
(17)D(J 2.18: Cho a > 0, b > Chitng minh: (a + bf +
Dau dang thiJc xky nao ? -A
\
Gidi Ap dung BDT Cosi tac6:
• a + b> lyfab =^ (a + bf > 4ab
a b ab in >
Cong (1) va (2) ve theo ve, ta dufcJc:
(1) (2)
(3) Lai ap dung BDT Cosi cho hai so": 4ab va ab Tac6: Aab + >2
ab \
Aab.—ab = (4)
'-A
Tir (3) va (4) suy ra: {a + b) + Dau ding thtfc xay <^ a = ^ =
>8 2.19: Cho a,b, c > va a + b + c = L Chtfng minh:
>64
a A c
Dau ding thiJc xay nao ? GiSi
Taco: + = + ^ + ^ = + + - + - (*) a a a a Mat khac ap dung BDT Cosi cho bo'n s6': 1,1,
32
Ta drfdc: ! + ! + - + £ > 4 / ^ a a V a Do dd tir (*)
• TuTdngtu":
a \a
l + i> 4
b ca 1
1 + - > 44 c
ab V
Nhan (1), (2), (3) ve theo ve ta dufdc:
(1) (2) (3) 1 +
a A A c
s
c
>64^ >64
'fee ca aZ? (dpcm)
Dau dang thiJc xay
a a
b b ^a^b = c = -1 = -1 = ^ = ^ c c
a+b+c=l
Bi 2.20: Cho a, b, c > va abc = Chifng minh: - + b + c a - + c + a b - + a + b c >27 Dau ding thiJc xay nao ?
(18)a be • - + c + a>3l
b
ca
1
• - + a + b>3l
c \ b
Nhan (1) (2), (3) ve'theo v e ' f dtfd^:
b
(1)
(2)
(3)
c
be ca ab a • b c
/ - \
- + ^ + c f l > iV^abc
a \b )
Mat khac theo gia thie't: abc =
Vay:
Dau dang thifc xay ra <f=^ a = ^ = c =
f l
- + fc + c - + c + a > 27 (dpcm)
\a } \b ) c ,
De 2.21: Cho a > b > Chiang minh: a +
' DS'u dang thiJc xay nao ? GiSi
(a-b).b ia-b).b
(BDT Cosi)
1
i(a-b).b
Dau d i n g thtfc xay <^
> (dpcm)
a = b = \
34
B6 2.22: Cho a > 0, b > Chu-ng minh: < / ^ ,
Gidi
0, b > Ap dung baft dang thiJc Cosi cho hai s6' du'dng, ta c6:
2\fja4b <4a+^ =^2i/ab <4a + yfb
2^b-j^<(^a + S ) ^ ^
4a^4b ^ '4a+4b
2 ^
4a + y[b < '4ab (dpcm).i
D d :
1) Cho a>\,b>\ Chtfng minh: a^Jb-X + b^a- \
2) Cho ba s6' a, b, c doi mot khac ChiJng minh: {a-Vbf , ib^-cf {c + af
{a-bf"^\b-cf • (c-a) 2 — >2 Gi§i
1) Ap dung bat dang thtfc C6si cho hai so' khong am, ta c6:
V r ^ = 7( Z j - l ) l < b-l + \= ^^a4b^\<^. (1) ab 2 ~
Chitng minh tu'dng tu" cflng c6 :
Cong (1) va (2) va' theo ve' ta diTcfc : a — <ab (dpcm)
b4^\<±. (2)
2
2) Da,t: r = ^'T = = 2-a —o o —c c — a
(19)2a 2b 2c Khid6:* (x + l)iy + l)(z + l) =
* (x-l)(y-l)(z-\) =
a — b b — c c — a %abc
' ia-b){b-c){c-a) lb Ic la
a — b b — c c — a Sabc
' ia-b){b-c)(c-a)' Dodd:
(X + \)(y + l)(z + \) = (x- l)(y - \)(z -1)
<!F^xyz + xy + xz + yz + x + y + z + 'i=xyz — xy — xz — yz + x + y + Z' ^ xy + xz + yz = - \)
Matkhactacd:
(x + y + zf>0^x^+y^ + z^+2{xy + xz + yz)>0 (2)"
(1) va(2) = ^ A : ' + / + Z' - >
= ^ x ^ + / + z ' > (a — b) {b — c) {c — a)
De 2.24: Vdi a >0, b > 0, c > Chiang minh cdcbat dang thiJc sau:
1) ±^^S>2b
c a
ab be ca ^ , , , 2) — + — + -—>a + b + c
c a b
2ab 2bc lea • 1) Ap dung bat ding thifc Cosi cho hai so' dtfdng ta cd
c a \ a ± , ^ ^ ^ + ^ > l b (dpcm) 36
2) Tacd: ^ + ^>2b c a ChiJng minh tu'dng tuf; ta cd:
1 >la
c b be ca ^ ^ — + — > c
(1) (2)
(3) (1) +(2) + (3): — + — + — + — + — + — > + 2a + 2c c a c b a b ~
ab be ca
+ — + — > a + Z7 + c.(dpcm) c a b
3) Tacd: {a + b){a-bf >0=i^{a^b)[{a^-ab^b')-ab
=^a^ +b^-ab{a + b)>0 ^a' +b^> ab{a + b)
>
lab ChiJng minh tu'dng tu" ta cung cd:
Ibe c'+a' lea (4) + (5) + (6) :
b' +c' ^ b + c
>
1
c + a
a'+b' , b'+c' c'+a' lab • + Ibc lea
> ^ (4), (5)
(6) > a + + c (dpcm) Be 2.25; Cho a>Q, b>O-Chitng minh: 3a' + lb' > 9ab^
Giii
Ap dung BDT Cosi cho ba so'khong am 3a';3b';4b\
(20)Tac6: 3a' + ^ ' +Ab' >^l]3a\3b\Ab\
^3a' +lb' >3ab\lj33A (1) Mat khac: IJ^JA > ^ 3 = (2) (1) va (2) ^ 3a' + 7b' > 9ab^ (dpcm)
§ BAT D A N G THUfC TRONG TAM G I A C Cho tam giac ABC Dat: BC = a; AC = b; AB = c
Khido: A a >
1) b>0
OO B
2)
\a-b\<c<a + b \b-c\<a<b + c \c-a\<b <c + a
Bi 2.26: Cho a, b, c la dai canh ciia mot tam giic ChiJng minh rkng: +b^ +c'^ < 2{ab + ba + ca)
GiSi Theo tinh chat tam giac ta c6:
0 < a < + c < a(Z? + c)
Q><b<a^-c^b'^ <b{a^-c) Q<c<a^-b=>c^ <c{a^-b)
C6ng v6' theo ve ba ding thtfc tren ta dufdc: a^ <2(a^ + fea + ca)
2.27: Cho a, b, c la dai ba canh ciia mot tam giac
Chiang minhr^ng: abc>(a + b-c)(b + c-a)(a + c-b) Giii
Tac6: a' >a^ -{b-cf ={a + b-c)(a + c-b)>0
(21)b"" >b^ -(a-cf =(a + b-c)(b + c-a)>0 '^c^ -(a-bf =ia-\-c-b)ib + c-a)>0 Nhan ve' theo ve'ba bat dang thiJc tren ta cd:
a'feV >ia + b-c)\a + c-b)\b + c-af =^ abc>ia + b- c){b + c - a){a -^c-b)
2.28:
1) Cho x > O v ^ y > O C h u ' n g m i n h r ^ n g : - + - > — ^ D i n g
X y x + y
thtJc xay nao ?
2) Trong tam giac ABC cd chu vi 2p = a + b + c (a, b, c la dai ba canh tam giac) ChiJng minh rang:
D i n g thu-c xay 1
• + > p—a p—b p—c
tam giac ABC la tam giac gi ?
a b c
1) T a c 6: i + i > - i - ^ « ^ > "
X y x-\-y xy x + y
'^{x-'ryf > x y
^{x-^yf -Axy>0 ^{x-yf>Q (diing) Vay: — + - > ^ Dau dang thiJc xay o x = y
X y x + y
a+b+c -a+b+c
2) p-a = a = >
^ ^ 2 V i a, b, c la ba canh cua mot tam giac nen b + c > a
=^'-a + b + c>0^p-a>0 Tu'dng tu* : p - b > 0; p - c >
40
Ap dung 1) ta cd:
p—a p—b p—a+p—b 2p—a—b c
Tu'dng
tu-Do dd
' >1
p-b p — c a' p-a p-c b
p-a p — b p — c
>
\a b c >
a b c
Dau xay <^
tam giac deu
p-a p-b p-c p — a = p — b
p-b = p-c^a=^b = c^ AABC la p—c=p—a
De 2.29: Cho a, b, c la dai ba canh cua mot tam giac vdi chu vi 2p
Chiang minh rang: (p - a){p - b){p -c)< abc
Gi^i
T^ a^ x * a + b + c b + c-a T a c o : * p-a = a =
2 ^ , a+b+c , a+c-b * p-b = b =
* p — c — a+b+c ^_a+b-c 2 Nhan (1), (2), (3) ve'theo veta du^dc:
(1)
(2)
(3)
{p-a){p-b){p-c)^ {a + b-c) (b + c-a) (a + c-b) 2 2 Mat khac: (a + b-c)(b + c-a)(a + c-b)< abc.( De 2.27 )
(22)Vay: {p - a){p - b){p - c) < abc
~8~ (dpcm)
2.30: a, b, c la dai cua mot tam giac Chitng minh rang:
<
a ^ c a c b
b c a c b a
Ta c6: V T =
GiSi
<^ a c b b c a c b a
a b' b c' c a]
+ +
b a, + c b, + a c)
1 b'-c'
ab be ca
abc c.(a^ -b^) + aib^ - ) + Z?(c' - a') Mat khac: c(a' -b^) + a(b^ - ) + b(c^ - a')
= c c'-b'
Do d6 :
= c(a' - ) + c(c' -b^) + a(b^ - ) + &(c' - a') = ( a ' - )(c - Z?) + ( c ' - 6')(c - a)
= ( a - c ) ( c- Z 7) ( a + c - c - ^ ) = (a-c)(c-b)ia-b)
(a-c)ic-b)(a-b) b.a.c
V T = <
abc abc V\:\a-c\<b;\c-b l< a;l a-b\<c
= (dpcm)
42
§4 PHUdNG PHAP LAM TRQI
Dilng tinh chat cua bat dang thtfc de du'a bat dang thrfc can chiJng minh ve dang c6 the tinh diTdc theo phrfdng phap tinh tong hffu han Gia su" can tinh tong: S„=U,+U^+U^+ + [/, + + / „
Ta phan tich: — a,^ — K h i d o :
S„ -a^) + (a^-a,) + (a,-a,) + + (a„- ) = - a „ + , •
2.31: Cho n la so' nguyen thoa n > ChiJng minh:
i + ^ + i + + _ J _ < ,
1.2 2.3 3.4 n(n + \)
T a c d : — = 1- 1.2
1 1 2.3
3.4 ~
1 1
(n — l)n n — 1 n 1 n(« + 1) n n + l
Cong ve' theo ve va ddn gian cac ding thiJc ta diTcJc:
1.2 2.3 • + • = - -n + l
(23)Do do: i + J , + + ^ < , , 1.2 2.3 n(n + l )
2.32: Cho n la so' nguyen thoa n>2. ChiJng minh: , , , n - - ^ + ^ + + -T< •
Giii
1 2^
1 < —
1.2
= - '
3'
1 < —
2.3 _ ~
1 1
S
_ 1 4^
1
S 1
< — 1
n (n — \)n n — l n
Cong ve theo ve' cac bat dang thiJc tren ta difdc: , , , 1 1 1 n-\
— + r r + + - r < l •
n n n
De 2.33: Cho n la so'nguyen du'cfng Chitng minh r^ng: 3n + l , , ^«
n
2 ( n - l ) S l - + r + ^ + - + - < - n
Giii
1 T a c o : +-1 + ^ + + ^ < + ^ + J- +
2 ' ' 1.2 2.3 {n-\)n
< + f1 ' 1 — f ^ M
+ + +
2 ; 3, rt — 1 n
T a l a i c o : l + 4r + i + - + -V>l • ^
2.3 3.4 < - i
n(n + l ) 44
1 r 1 1
+ + +
2 ' 3, 3 4, ,« « + i > +
> l + i - ' ^ 3n + l
V a y :
2 n + n + ( « + l )
2 ( n - l ) 1' 3' .2 — • n
D^ :
1) Cho k > Chitng minh: < 1
(2Jt + l ) ' Ik 2k + 2) Cho so' nguyen dicing n > Chitng minh :
9 25 49
1
(2n + l ) ' ' ^ '
(1)
1) T a c d (1) <^ < 2 {2k + \f 2k(2k + 2)
4^ (2;t + If > 2k{2k + 2) <^ ^ ' + 4;k + > Ak^ + Ak
<^ > Die 11 luon diing
V a y :
<
(2^ + 1)' 2k 2k +
2) A p dung cau ta c6:
2 1 V d i k = thi <
-9 k = t h i A< i _ i ,
25 k = M A < i _ i
49
k = n t h i
<
1 (2n + l ) ' 2n 2n +
(24)C6ng vd" theo v<S' n ba't ddng thitc tr^n ta duTdc: 2
5+ + ? ^ - + - ^ '
1
(2« + l ) ' ^ 2n + ' ^ '
Do 66:
9 25 49
1
<- (2n + \f
2.35: 1) Cho so'nguyen k>\. Chitng minh:
< 1
2) Cho s6' nguyen n > Chiang minh:
• + 1 + + • 3^2 4^/3 (n + l)V^
• <
(1)
(2)
1) Tacd (1) 4=^
1
<
Giii
(k + l) <
( ^ + ) ^
4k + \-yJk
^ / ) t ^ / ^ f l
1
<
yfk + l
^^/k < yfk + l. Bat dang thiJc lu6n diing
Vay : <
(A: + 1)V^ 2) Ap diing cau 1) ta c6:
k = 1 t h i - <
2
1
4k V T H ,
k = 2 thi
k = thi
< 1
• <
172
1
4V3 V4,
46
k = n thi <
Cong ve theo ve cac ba't dang thitc tren ta du-dc:
Vay :
2 + ^
<2
- + •
2 ^ "• (rt + l)V^ <
<
(25)CHlTOfNG III: SO HOC I.TINHCHIA HEX
1) Dinh nghia: Cho a, b la cac so nguyen Ta noi a chia bet cho b neu ton tai mot so nguyen q cho:
a = bq
Ki hieu: a:b Khi ta cung noi b chia het a va ki hidu: b|a 2) Tinh chat:
a) Neu: a chia het cho b va b chia he't cho c thi: a chia he't cho c
b) Ne'u: a, b cung chia he't cho c thi: a + b va a - b chia he't cho c
c) Ne'u: a chia he't cho c hay b chia het cho c thi: a b chia he't cho c
3) Dau hieu chia he't:
a) Dau hieu chia he't cho 2: chi? so tan cung la chan ; b) Dau hieu chia he't cho (cho 9): tong cac chi? so chia
he't cho (cho 9)
c) Dau hieu chia he't cho (cho 8): hai (ba) so' cuo'i chia he't cho (cho 8)
d) Dau hieu chia he't cho : chu" so' tSn ciing 1^ hoSc e) Dau hieu chia he't cho 11 • hieu giffa tdng cac chiJ so d
vi tri chan va tdng cac chff so'd vi tri le chia he't cho 11
II SO NGUYEN TO
1 Mot so nguyen du'dng du'cJc ggi la so nguyen to ne'u no chi chia he't cho va cho chinh no
• Nhtf vay theo dinh nghia nay, so' la so nguyen to 48
Tuy nhi^n cic s6' nguyen t<5' tham gia l^m thiYa s6' ciia mot tich, ngtfdi ta qui rfdc khong ki d6'n so' 2 S6' nguyen dufdng kh6ng la s6' nguySn t6' duTcJc goi Ik hdp
s6'
3 Gia siJ so' nguyen difdng N diTdc phan tich thilfa s6' nguy6n t6':
N = a'".b\c'
Khi dd tdng cac u'dc s6'khdc cila N Ik: (m + l)(« + l)(^ + l)
III U(5C CHUNG L(5N NHAT - BOI CHUNG NH6 NHXT
1) Binh nghia 1: Udc chung Idn nhS't (l/CLN) ciia hai s6' a b khong ddng thdi bkng la s6' nguySn Idn nhat chia ha't ca a vkb
Kihi6u: (a ,b) d^ chi UCLN cila a vk b
2) Binh nghia 2: c^c s6' nguyen a, b diTdc goi Ik nguySn t6' cing n^'u: (a, b) =
3) Tinh chat: b) = \a + h\\= (\a + h\h) 4) Cach tim UCLN cua a vk b:
• Phan tich a va b thknh thiTa s6' nguyen t6' • Lay tich cac thu'a s6' chung vdi s6' mu be nhat 5) Binh nghia 3: Boi chung nhd nhat (BCNN) cua hai s6' a va b
la so' h6 nhat chia he't cho ca a va b Kijiieu: [a;^] de chi BCNr cua a va b 6) Cich tim BCNN cua a va b:
• Phan tich a vk b t'lii'a so' nguyen t6'
• Lay tich cac thil'a s6' chung va rieng vdi s6' mu Idn nha't
IV.DONGDUTHLTC
(26)ta duTdc cilng dif so'
Khi 66, ta k i hien: a = b (mod m)
2) a = b(mod m)-^ a — b + km vdi k la so' nguy6n 3) Tinh chat:
* a = a(modm)
* a = ^(mod m)=> b = a(mod m)
a = b{Kodm)
b = c(modm)
a = ^(modm)
a = c(modm)
c = c/(mod m)
* a = Z>(mod m) =^
a±c = b± c(mod m)
ac = b.d{modm) na = nb(modm) a" =b''(modm)
V SO C H I N H PHUdNG
1) Binh nghia: Neu a la so' nguyen thi a^dmc goi la so' chinh phifcfng
2) Tinh chat: Tan cung cua so'chinh phu'cfng 'a 0, 1, 4, 5, 6,
V I N G U Y E N L Y D I R I C H L E T
Khong the nho't chit tho vao chiec long, cho moi long c6 khong qua hai chit tho
* Nhu" vay neu nho't chit tho vao chiec long thi it nha't phai c6 mot chiec long c6 tu" chu tho trd len
V I I NHJ T H l f C NEWTON
( a +b f = cy + cy-'b+cy-^b^+ +cy
50
De 3.1: Chifng minh r^ng vdi moi s6'tur nhien n > ta c6: 3'"+' + « - ;
Gidi
Ta C O: ' " + ' + 40« - 27 = 3^3'" + 40n - 27 = 27.9" + 40n -21 = 27.(8 +1)" + 40n - 27
= 27 (8" + Cl^"-' + + C"-^8' + C;-' +1) + 40A7, - 27
= 27 (8" + C' 8"^' + + Cr'.S') + 27.Cr' + 27 + 40n - 27
= 27.8'.^ + 27.8rt + 40n - 27.64.^ + 256n = 64(27^ + An) Vay: ' " + ' + n - ;
Ghi chu: C ° = l ;Cr'=n Tong quat: =
A:!(n -A:)!
Trong do: n! = 1.2.3 n (vdi n6 iV*) va 0! =
3.2: Chifng minh rang tong 2p + so tu" nhien lien tie'p chia he't cho 2p +
Gidi
Gia su" 2p + s6' tuT nhian lien tie'p la: k; k + 1; k + , , k + 2p Khi d6: S = k + (k + 1) + (k + 2) + +(k + 2p)
= A: + ^ + A:+ + /: + ( l + + + + 2p) 2p+l so'
= {2p + l)k + 2p 2p + l
=^i2p + l)k + p{2p +1) = (2/7 + \){k + p) Vay: S\2p + \
Ghi chit: V d i moi so'tif nhien n ta cd: l + + + + n = «(n + l )
(27)3.3: Cho n e N'. Tim UCLN cua hai s6': 2n + va n + Gi§i
Ap dung tinh chat: (a,b)= {\a±b\,b)^{a,\a± b l) Ta c6: (2n + 3;n + 7) = (l n - l,n + 7) = (l n - 1,11)
Nhtfng 11 Ik so' nguyen to' nen so' can tim la hoSc 11 Khi do: * Khi In - 41 la bpi so cvia 11 thi:
UCLN(2n + 3;n + 7) = 11
* Khi In - 41 khong la boi so' cua 11 thi: UCLN(2n + 3;n + ) =
3.4: Mot so' nguy6n du'Png N c6 dung 12 tfdc s6' (du'dng) khac ke ca chinh n6 va 1, nhiTng chi c6 ba tfdc so' nguyen t6' khac Gia su" tong cua cac ufdc so' nguyen to' la 20, tinh gia tri nho nhat C O the cd ciia N
Gi^i
Goi cdcirdcnguy6n to'ciia so'N la p, q, r va p < q < r
f^ = 5;r = 13 q = l;r = U
[A^=2".7Mr
V d i a,b,c e N va (a + l)(b + l)(c + 1) = 12 T a c d : 12 = 2.2.3
Do d N c thela : 2.5.13^2.5^13 ; 2^5.13 ; 2.7.11^ ; l l l ; 2^7.11 N nho nhat nen N - 2^5.13 = 260
3.5: Chitng minh r^ng c6 s6' nguyen drfdng chi chiJa cac chi? s6' va chi? so va so chia het cho 1999
52
Xet 2000 so'nguyen du-png sau: 1; 11; 111; ; H L ^
2000 chuso'X Cac s6'tr^n chia cho 1999, ton tai it nhat hai sd'chia cho 1999 cd cilng so'du" (nguyen ly Dirichlet) Goi sd'dd la :
11 • 1. va 11 (2000 > m > « > 1)
m chusd'l n chusd'l
» 1 - ,11 chia het cho 1999 m chusd'l n chusd'l
^ 11 100 chia h^'t cho 1999 (dpcm)
m—n chiisd'i n chuso'O
Bi 3.6: Cho n la so' nguyen diTdng Idn hcfn ChiJng minh n" + + la mot hdp so'
Gidi Tacd: n ' + n ' + l - n ' + n ' + l - n '
= (n^ + lf-n^ = (n^+n + l)(rt' - « +1) V d i n > l , t a c d : n ^ + n + ^ va n ^ - n + 15^1
Vay: + n ' + l l a m p t h d p s o
Bi 3.7: Tim s6'nguy6n dtfdng n de: -n^ +n-l la s6'nguyen to
Gi§i '
Tacd: A = n'-n^ +n-l = n ' ( n - l ) + ( « - l ) - ( « - l ) ( n ' + ) • Khi n = 1, A = 0: khdng la so' nguyen t6'
• Khi n = 2, A = 5: Ik so' nguyen t6'
• K h i n > : T a c d : n - l > va n ' + l > 0 n a n A l a hdp s6'
Tdm lai: Khi n = thi -n^ + n-\a so'nguyen to'
(28)3.8: Cho p va + la hai s6' nguyen t6' Chitng minh: + cung la s6' nguy6n to'
Gi^i
Dat: p = m + r ( v d i meN';r = 0;1;2 )
K h i d d : + = (3m + r ) ' + - 3(3m' + 2mr) + r^+2
• V d i r = : T a c d p = 3m
V i p la so' nguyen to' nen m =
=^ p = =^ + = 29 + la so nguyen to
• V d i r = 1: Ta CO
+2 = 3(3m' + 2m) + + = 3(3m' + 2m + ) V i : m G A^* =^ 3m^ + 2m + >
=^ + khdng la so' nguyen t6'
D i e u trai v d i gia thie't
V a y khong xay tru'dng hdp r =
• V d i r = : T a c / ? ' + = ( m ' + m ) + +
= ( m ' + m + 2)
- , V i : 3m^ + 4m + > nen + la hdp so
D i e u trai v d i gia thie't
vay cung khong xay tru-dng hdp r =
T d m l a i : K h i p va /? V la so' nguyen to t h i : p =
K h i d d : + = 29 cung la so' nguyen to' (dpcm)
3.9: ChiJng minh rang:
1) V d i m o i so'nguyen du'dng n, phan so'sau la to'i gian: ^ ^ " ^ ^ 14n +
2) N^ l a s o v o t i
54
Giii
1) Ta se chtfng n i n h : 21n + va 14n + la hai so' nguyen to' ciing
^ That vay: ne'u goi d(d > 1) la U C L N cua hai so' tren i i n + = pd
• va 14n + = qd ^2)
trong dd p, q la nhifng sd'nguyen, du'dng
( l ) - ( ) : n + l = ( p - q ) d = ^ « + = ( / ; - ) J (3)
( l ) - ( ) : l = p.d-3(p-q)d = (3q-2p).d=^d = 3q-2p = l
V a y : 21n + va 14n + la hai so' nguyen to' cung
td-c la: i l ^ i ± l la phan so t o i gian 14n +
2) Ta chiJng minh bang phan chiJng
G i a i su- rang V2 e g v d i ^ = ^ (viet du^di dang to'i gian nghia la
1
p va q la hai so' nguyen to' ciing nhau)
T a c d 4- = = ^ / =2q^
^ p^ la so'nguyen chan =^ p Ik so'nguyen chSn
^ p = 2m ( v d i m G Z ) =^ 4m^ = 2^^ =^ = 2m^
q^ la so'nguyan chan =^ ^ la so'nguydn c h f n
=^q = 2k
V i p = m va q = 2k nen chiing khong nguyen td' cung (cd tfdc so' chung la 2) D i e u mau thuan
V a y b ^ t b u d c yf2 l a s o v o t y
t>i 3.10: Chu-ng minh rllng: 3" = - l ( m o d l O ) K h i \'k chi k h i 3"^' = - l ( m o d l O )
(29)Giii
Tac6: r =-l(modlO) 3" -10A:-1 <^3"+' =(10/:-l).81.<i4.3''^'' =810A;-81 ^3"+' =(8U-8).10-l.<f::^3"+'' =10^-1 <!=J^3"+'* =-l(modlO)
3.11: ChiJng minh rling vdi moi s6' nguyen difdng ta c6: 1376" =1376(mod 2000)
Ta dung phufdng phap qui nap de chiJng minh
• Vdi n = 1: De thay: 1376^ = 1376(mod 2000) Ttfc menh dd dung n =
• Gia su" menh de ddng tdi n = k Ttfc la: 1376* =1376(mod2000)
=^ 1376*+' = 1376'(mod 2000) (1) Matkhac: 1376^=1893376 -1892000 + 1376
= 946.2000 +1376 =^ 1376' = 1376(mod 2000) (2) Tit (1) Yk (2) 1376*+' = 1376(mod2000)
Nghia la mdnh de diing tdi n = k +
Vay vdi moi s6'nguyen difdng n, ta cd: 1376" = 1376(mod2000) Bi 3.12: ChiJng minh r^ng s6' jc = 10^" +10'" +10^" +1 chia het cho 7, 11 va 13 neu n le Tim so' dif cua phdp chia x Ian liTdt cho 7, l l , v a l l l
Giii .x = 10'"+10'"+10'"+l
TH,: Vdinle.tacd: * 10'" +1 = (10')" -(-1)" chia het cho
56
10'-(-1) va: 10'-(-1)-1001 =
7.11.13-Do d6: 10'" + chia het cho ; 11 ; 13
Matkhdc: ;c = 10'"(10'"+l) + 10'"+l = (10'"+l)(10'"+l) Do dd: X chia het cho ; 11 ; 13
11^2: Vdi n chan
10'"+l = [lO'"-(-l)"] + chiacho7 ; 11 ; 13dir2 10'"+l = [lO'"-(-l)'"] + chiacho7;ll ;13dir2 * Do dd X chia cho ; 11 ; 13 deu cd sd'diT la
* Ngokiratacd: 10'"+1 = [(10')"-l"] +
Vi: (10')" - " chia ha't cho (10' -1) nhiTng: 10' - = 999 = 3.111 chia he't cho 111
Do dd 10'" +1 chia cho 111 du" vdi moi neN Tu-dng ttf 10'" +1 chia cho 111 dtf vdi moi n e iV Suy x chia cho 111 dtf vdi moi neN
Tom lai: x chia cho ; 11 ; 13 duT n^'u n le ; dtf na'u n chan X chia cho 111 du" vdi moi neN
Bi 3.13: Cho so' nguyen n > ChiJng minh ring: n" -«'+«-1 chia he't cho (n-lf
Giii
Nhan x^t rkng vdi n = thi : n" -n^ +n-l= (n-\f= nan bai , toan hien nhian dung
Bay gid ta xet n > 2:
Tacd: n"-n" + n - l = («""' -l)n' + ( n - l )
= (n - l)(n"-' + n"~' + + « + l)n' +(«-!) = (n - l)(n"-'+ n"-'+ + ) + (n -1)
(30)=l + k^(n-l) (vdi k^=n + l)
n"^' = + k„_^ ( « - ! ) ( vdi k„.i = n"-^+ +n +1 ) Cong ve theo ve ta c6:
+ + + = (n - ) + (k, + + ).(n - ) = ( n ^ l ) ( l + k i + + k , , i ) nen n"~' + + + chia he't cho n -
Do tu" (1) ta s«y rang n" -n^ +n-\a het cho (n - ) l
De^ 3.14: Xet ba so t\i nhien a, b, c th6a he thiJc: ^b^+c\ ChiJng minh r i n g neu a, b, c nguyen to' cung thi a la so' le va
trong hai so' b, c c6 mot so' le va mot so' chfn
Giii
Gia su" a chfn
V i a, b, c nguyen to' cung nen b, c le • a chan chia he't cho
• h,c le =>b + c chia he't cho va be khong chia he't cho ^ib + cf - 2bc khong chia he't cho
=^b'^ khong chia he't cho Nhtfng theo giai thie't: = ZP^ + c^ V6 l i !
Vay a le ma a' = + =^ c6 mot sd' le va m6t so' chJn ^b,c CO mot so' le va mot s6' chfn (dpcm)
Y)i 1.15: Chu-ng minh rling tong cac binh phu-Ong cua ba so' nguyen hen tie'p khong the la lap phu'dng cua mot so' nguyen dtfcfng
Giii
Tru'dc he't ta chitng minh neu n e Z thi: • chia cho dtf hoac
58
* chia cho dtf hoac hoac
That vay: Dat n = 3m + r{n,m eZ;r=- 0;±1) * = (3m + rf - 9m' + 6mr + r '
r ' = ;1 nen n'chiacho du-Ohoac (1) • * ==(3m + r)^ = m ^ + m V + m r ' + r \
= ;1 ; - l nen chia cho du" hoac hoac (2) * Goi ba so nguyen hen tie'p Ian lu-dt la a - ; a ; a +
Ta CO (a - ) ' + a' + (a + ) ' = a' - 2a + + a' + a' + 2a + = a ' +
Tu" (1) ta CO 3a' + chia cho drf hoac ma chia cho du" hoac hoac (tu" (2))
Do 3a' + khong the la lap phu'dng cua mot so'nguyan drfdng
3.16: ChiJng minh rang tich bo'n so'nguyen du-dng l i ^ n tie'p khong the la so' chinh phu'dng
Giii
Giai siJ ton tai so' nguyen m de:
m ' = nin + l)(n + 2)(n + 3) <^ m ' = ( n ' + 3n)(n' + 3n + 2) = k{k + 2) (vdi k = n ' + 3n )
Tacd: y t ' < ) t ( ^ + ) < ( ^ + l ) '
Dieu mau thuan vdi k{k + 2) = m ' TO suy dieu phai chifng minh
3.17: Tim so' nguyen n cho: n' + 2n' + 2n^ +n + l la mot so chinh phu'dng
(31)Giii
Theo de bai ta c6: n* + 2n^ + 2n^ + n + = (vdi m e Z ) ^ A{n' + In" + 2n^ + rt + 7) = Am"
4=^ ( « ' ' + 8n' + n N 4n +1) + 27 = W <i=^ (2n^ + 2n +1)^ - 4m^ = -
^ {In' + 2n + - 2m)(2n^ + 2n + + 2m) = - Do c6 cac trifdng hdp sau day xay ra:
2 n ^ + n + l - / n = l « ^ + n + l + 2m = - '
2 n ^ + n + l - m " = - l !)•
2)
3)
4)
5)
2n^+2n + l + 2m = 27
2 n H n + l- 2/ n = - l « = —3 /la}' n =
-2n^+2n + l - m = 2n^ +2n + \+2m = -9 2n^ + 2rt + l- m = -
2/i^ + 2n + l + 2m =
2 n ^ + « + l- m =
2n^+2n + l + 2/n = -
4n^ + 4n + 28 = 0 (vdnghiem) « ' + n + l + 2m = -
2n^+2rt + l - m = - l
/n = m =
n = -
n =
2n^+2Ai + l- m =
4n^ + 4n + = (vo nghiem) 2n^+2n + l- m = -
+ n 4-1 = 0 {vo nghiem) n ^ + n + l - m = n ' + n - l =
V d i + n - = <!:^ n = i ( - ± ^/5): khong la so nguyan (loai)
6)
7)
2n^ + n + l - m = - 2n^ + n + H-2m = 2n^ + n + l - w = 27 2n^ +2n + \ 2m = -l
2 « ' + n + l - m = - +n + — {vo nghiem) n ^ + « + l - m = 27 n ' + n - =
n =
m =
n- -
60
8)
2 n ^ + n + l - m = : : - n ^ + n + l-|-2m = l
2 n ^ + n + l - m = - n ^ + n - | - = : 0 {vo nghiem) , Vay cac s6'c^n tlm la: n = 2, n = -3
f)i 3.18: Tim ba so' t\I nhien m, n, k d6i mot khac cho J- + i + - la mot so' tu" nhien
m n
Ta CO the gia su" r^ng m < n < k
Dat / = — + - + - vdi i la s6' tu* nhian m n k
V i m < n < k - >i>i=»/ = l + l + i < A m n k m n k m
3
= ^ I < (vi m > =j> — < ) =j> / = hoac i = m
* V d i i = thi c6 < — =^m < =j> m = hoac m = m
• V d i i = m = thi khdng th6a
m n k [ I n k ,
• V d i i = 1, m = thi dieu kien
m n k n k n k 2
Suy ra —>-=^n>4 n
vt - > —
n k
Ngoai = m < n < =^ n = va dd:
/ = - + - + - < ^ ^ = m n k
(32)* V d i i = tl>i c6 I < — =4> m < - =4> m = m
' L t i c d d d i l u kientrd thanh: = + - + ^ = ^ - + ^ = !
n k n k
M a m = l < n < k n e n n > , A : > - ^ - + | < ^ + ^ - < l n A: Do d6 khong thoa dieu kien neu tren
Ket luan: m = 2, n = 3, k = la ba so' can tim
3.19: T i m cac so' nguyen difclng m, n cho: 2m + chia he't cho n va 2n + chia he't cho m
GiSi
Trirdc he't ta xet triTdng hdp \
Ta c6: m < 2n + < 2m + (vi 2n + 11a boi so' cua m) 2n + = m hay 2n + \ 2m hay 2n + \ 3m
(vi 4m > 2m + nen kh6ng the nhan 2n + = m , ) • 2n + = m:
Ta c6: 2m + = 2(2n + 1) + = 4n + chia he't cho n nen suy chia h6't cho n, v i the phai c6: n = hoac n =
V d i n = t h i m = V d i n = t h i m = • 2n + = 2m:
Ta CO 2m + = 2n + chia ha't cho n nen suy chia he't cho n va do: n = hay n =
3
V d i n = thi m = - (loai) V d i n = thi fn = ^ (loai)
• 2n + = 3m
62
Ta cd 2m + > 2n +
2m + > 3m =^ m < =^ m = V d i m = t h i n =
Do ta thu du-dc: (n = 1, m = 3),(n = 3, m = 7), (m = n = 1) jsfhan xet rang vai tro ciia m va n nhu* de bki, vay
1 < m < « ta tim them du-dc: (m = 1, n = 3), (m = 3, n = 7)
D6 3.20: M o t so' chinh phu-dng cd dang abed. Bie't ab-cd = l. Hay tim so' a'y
Dat abed = vdi a, b, c, d la cac chff s6' Ta phai cd a^O,
ngoaira, do ab-cd = l n€n : e^O
T a c d : = 0 a i + ^ = 0 Q + l ) + ^
Suyra: - 0 = 1 ^ - ^ ( n + ) ( « - ) = 1 ^
V i n^cd bo'nchu'so'nen n < 100, dd n + 10 = 101, hay n = 91 Vay 8281
Bi 3.21: T i m so' tu* nhien n = ab cd hai chu" so' cho:
n + ab = (a + hf
T a c d : n = ab = \Oa + b
Theo gia thiet: n + ab = ( a +b )"
^I0a + b + ab = a^ +b^ +2ab ^b^ +ia-\)b + a(a - ) =
^b^ +(a-\)b = a(lO-a) (1)
(33)Ta lai bi^'t r i n g (x + yf > 4xy
n€n (a + ( - a ) ' ) > a ( - a) = i- 0 > a ( - a )
= ^ a ( - a ) < (2) TO (1) va (2)suy ra:
+(a-\)b<25^b^ <25 (•v\(a-\)b>0) = ^ ^ < , d o d b = 0, 1,2, 3,4,5 * Neu b = 0: (1) a = : kh6ng phu hdp
* N e u b = l : ( l ) = ^ l + a - l = a - a '
=^ - a = a = (vi phai c6 a>l) * N e u b = 2: (1) + a - = l O a - a '
= ^ a ' - a + = = ^ a = 4±N/ l (loai)
* N a u b = : ( l ) = ^ + ( a - l ) = a - a '
= » a ^ - a + = = ^ a = l hay a =
* N e u b = 4: ( l ) = ^ + ( a - l ) = a - a ' =^ - 6a + = (v6 nghiem) * N e u b = : ( l ) - » + ( a - l ) = a - a '
I =^ a ^ - a + 20 = 0(v6 nghidm)
Ket ludn: n = 91, n = 13, n = 63 la cac so' can tim
64
CHL/CfNGIV: GIA T R I L6N NHAT
VA GIA T R I NHO NHAT CUA HAM SO
Neu CO hang so'M cho:
N6u CO h i n g s6'm cho:
I DINH NGHIA. Cho ham so' y = f(x) x^c dinh vdi x e D
/ W < M , V ; c e D
3x^eD:f{x,) = M thi M la gia tri Idn nhat (GTLN) cua f(x)
K i h i e u : M = max f{x)
'f(x)>m,WxeD 3x,eD:f(x,) = m thi m la gia tri nho nha't (GTNN) cua f(x)
K i hieu: m = f{x)
Ghi chu: Tap xac dinh D la tap cac gia tri x cho f(x)
CO nghia
II C A C H T I M G T L N VA GTNN CUA H A M S d
1) Loai 1: Dilng tinh chat: |A| > A Dau "=" xay
2) Loai 2: Gia su' A, B la cac hang so, B > va g(x) >
Cho: f{x) = A^ B
Khi do: * f(x) Idn nha't ^ g(x) nho nhat * f(x) nho nhat ^ g(x) Idn nhS't
B Cho: f(x) =
A-g(x)
(34)3) Loai 3: Dung dieu kien c6 nghiem cua phtfOng trinh bac hai
Ghi chu: Cho ham so y = f(x) xac dinh tren tap D Neu ph^dng trinh y = f{x)c6 nghiem xeD ^ a<y<b
thi f(x) = a va max f{x)^b 4) Loai 4: Diang cac linh cha't cua bat ding thiJc
4.1: TimGTNN cua : j = yjx-l-lyf^^ + ^Jx + l-6y[7^ Tac6: y = y]x-\-2y/x-2 +^lx + l-6yf7^
^ y = ^(x-2)-2yfI^ + \+yl{x-2)-6sf^ + • Di^ukien: A:>2 Khidd:
y = yJx-2-l + ylx-2-3
^y = ylx-2-\ 3-^x-2 >i^yJx-2-\) + {3-^x-2 Dau "=" xay ^ yJx~2-\>0 3-ylx-2>0
^ l< V x- < < ^ l < x - <
^ < ;c < 11 (thoa dieu kien) Vay: y =
Dat dtfcfc khi: < A: < H
66
p i 4.2: Cho M = yjx + 4sjx -4 + ^x-A^x-A Tim x de M nho nhS't va xac dinh gia tri nho nha't cua M
Gi^i
Dau "=" xay <^
Tac6: M = ^jx + Asfx^ + yjx-Ayjx - A
^ M = 4{X-A) + A^X-A^A + SJ{X-A)-A4I^^A
M = ^ ( V^ ^ + ) ' +^^(V7r^-2)'
Dieu kien: x > Khidd:
y; c- + 2 + V ; c- -
V^ - + + 2- V x- 4 > ( V x- + ] + f2- V ; c-4W4
V A: - + >
2_ V ; c-4 >
4=^0< V;c-4 <4<i=>0< A:-4<4
<^ < ;c < (thoa dieu kien) Vay 4 < A: < thi M dat gia tri nho nhat va M =
4.3: Tim GTNN cua ham so: y = + -4;c + Giii
Td c6: y = yfx^ + ^x^ -4;c +
4^ y^ \[x^ + ^[x-2'f (hilm so'xac dinh vdi moi x) ^y = + x-2
Da'u "=" xay
< ^ y = : A : + 2- A : > A : + (2- A : ) =
;c>0
^Q<x<2 2-x>0
Vay y = dat dtfdc 0 < A: < 2,
(35)4.4: Tim GTNN cua bieu tMc: A =
2 + yj2x -x^ +l'
Ta c6: A =
Gi^i
2 + V x- ; c ' + 7 + V- ( ^ - l ) ' +
D i l u kien: ^(x-lf +S>0 ^ (x-if <
4=^-2 V2 < X - < ^ 4^ - 2V2 < A: < + v ^
Khi d6: A nho nhat ^ + ^-(x-l)^+S Idn nhat
^ -(x-2)' + Idn nhat ^ (x-lf = ^ x = 1, Vay khi X = thi A dat GTNN va
3
minA = , = j=
2 + V- ' + + 2V2 Bi 4.5: Cho M =
GTNN c u a M
x^-2x l_
x'+\ \ ^x + 2 l- V ^ + 2,
Tim
Dieu kien xac dinh cua M la:
Tacd: M = x{x-2)
Gi^i
A: + >
VJC + ^
A: > -
(;C + ) (A:'-X + 1) • l - ( x + 2)
4^ - ) _ Jc(jc- 2)-(.y'-;c + l ) {x + Viix"-x + \) x^\~ + -;c + l )
68
-(;c4-l) -
(jc + l ) ( ; c ^ - x + l ) x'^-x^\ X —
Vay M nho nha't
-^2 Idn nhat
X
n
X
H— dat gia tri nho nhat
4
1
4=> = < ^ =
-2 -2
1
Vay X = - thl min M =
•
D l 4.6: Tim GTNN va GTLN cua ham so: y = x" +3x-\ x^ +2X-V5'
Gi^i
Ta cd y xac dinh vdi moi x vi + 2;c + = la v6 nghiem jc' + ; c - l
Tacd: y =
x" -^2x^-5 ^ x'y + 2A;>' + 5y = A;' + 3A: -
^ ^ ^ ( j - l ) ; * : ' + ( y - ) x + 5>' + l - (*) • V d i y = 1: (*) <^ - X + = ^ X =
(phu'dng trinh cd nghiem) (1) • V d i } ' : ( * ) CO nghiem khi:
A > 0 4^ ( j - ) ' - 4(3; - I X S j +1) >
^ / -12>' + 9- ( / - 4> ' - l) >
4=^16/ - 4> ' - <
2-2V53 + 2N/53
4=> < J <
(36)1 - V53 ^ ^ + 753 ,
^ < > ' < r ( y ^ l ) (2) o o
TO (1) va (2) suy ra: (*) c6 nghiem khi: ^ ^ <y< 1-V53 + V53
Vay: y=^ ,max y =
8
De 4.7: TiiriGTNN va GTLN cu& y =
x'-5x + l
Wi x^ - 5x + l ^0 v6 nghiem nen y xac dinh vdi moi x x^
Tac6:y = — — ^ y(x^-5x + l) = x^ x^-5x + l
^(y- \)x^ -5yx + ly = (*)
V d i y = l : (*) <s=^-5x + = ^ ;c = - ( p t c6 nghiem) (1; V d i y ^ 1: (*) CO nghiem khi:
A > <^ ( - ^ ) ' - 4iy - l)(7y) > < ^ / - ' + y > 4^ - / - + > ' >
^0<y<=j {y^\) 28 TO (1) va (2) suy ra: (*) c6 nghiem khi: < y < y
28 Vay: y = va max y = —
Dd4.8: Tim GTNN cua ham so: y = -^'^^ x'+2x^+\
70
Giii
x" +2x^+\ 'x^ + I f > nen y xac dinh vdi moi x x' +2x^+\)-(x^+\) + \
cd: y
x'+2x'+\
(x'+\f (x'+\f =
1
1
x^+l '
at t =
x ' + l
{hi y = t^ -t + \^y =
/
\2
1
Xet ? = - ^ + = 4=^ jc^ = <^ jc = ± ;
Vay ta cd khi ;c = ± thi f = - / - - i - = <s^ y = - • 2 ^
3 Vay y dat gia tri nho nha't la: y = —
4
4.9: Tim gia tri nho nha't cua ham so': y = x^ + \ -
X
vdi x^O
1
X X
+ 5,
Dat t = x thi t' = x'+\-2^x'+-l- = t^+2
X X X
Tacd: y = r' + - / + = / ' - ? + = ( / - l f + > Xet t = \ x - - ^ \ x^-x-\ 0^x = -(\±^f5]
X 2^ I
Vay = - ( l ± thi y = nen y dat gia tri nho nha't la: y =
4.10: Cho y = (x l)(x 2)(x + 4)(x + 5) Tim x de y dat gid tri nhd nha't Xdc dinh y
(37)GiSi
Tac6: y ^[(x-l){x + 4)].[{x-2)(x + 5)] ^(x^ +3x-4Xx^ +3x-\0) Dat t^x^ +3x-4^ x + -3
2
\2 _25 ^ _ 4 ~
thitadifdc: = r ( r - ) = - r ^y = {t-3f-9
Khi t = 3^x^+3x-4 = ^x^ + x - = jc = i( - ± 73?)
Vay x = - ( - ± thi y dat gia tri nho nhat y = -
4.11: Cho hai so' thiTc x, y thoa dieu kien x^ =1 Tim gia tri Idn nhat va gid tri nho nha't ciia x + y
Giii
Ta c6 (x + yf + (x - yf > (x + yf
^ 2(x^ +y^)>(x + yf (Dau " = " xay <^ x = y )
Ma x'+Z =1
Do do (x + yf <2^-y/2<x + y < ^ • Y6ix + y<^j2
x = y Dau " = " xay ^
Vay X + y dat gia tri Idn nhat la
x = y =
x-^y = ^
Vdi x + y>-42
Dau " = " xay x = y Vay X + y dat gia tri nho nhat la
<^ x =
y^-72
p d 4.12: Tim gia tri Idn nha't cua y = \x\.yj4-x^
GiSi
TAC6: y^ \x\.M-x^ = 4x^.^4 - x^ = y]x\4-x^) Xac dinh vdi -2 < x <
Ta CO x^ +(4-x^)> 2^x\4-x)- (BDT Cosi)
^ 4 > 27]^^(4^^ <^ > 7( ^ ^ ^2>y
D a u " = " x a y r a ^x~^4-x^ ^ x" = 2^ x = ± ^ Vay max y = dat du-dc ;c = ±sl2
DC- 4.13: Cho y = ^Jx^ +x + l + yjx^ -x + \ Tim x de y dat gia tri nho nha't Xac dinh y
V i x^+x + l =
x' -x + l^
\2
n
X
2
Giii
4
+ — > n6n y xac dinh vdi moi x
4
Dung ba't dang thiJc Cosi cho hai so'khong am a, b: a + b>2s/ab
Taco:
>2^{x'+lf-x'
(38)D d 4.14: Cho y = — T i m x de y dat gia t r i Idn nhat Xac dinh
X + max y
Gi§i
Ta biet rang + > lab nen suy ra: / + = {x'^f + ' > lx\
Ix" ^ >
X e t ^ = ^X' - 2A: ' + =
1 Do k h i x = ±.\i max y = —
2
4.15: Cho X > va y = x^^ - + T i m gia t r i nho nhat ciia y
Gi^i T a c d : y = ( x ' ' - ) - ( x ^ - l ) +
= x ' ( x " " - ) - ( x ' - ) + = x ' [ ( x ' ) ' - ] - ( x ' - ) + :
= x ' ( x ' - l ) [ ( x ^ ) ' + ( x ' ) ' + x ' + l ] - ( x ^ - l ) + ^
= ( x ^ - l ) ( x ^ + x ' ^ + x ' ° + x = - ) +
V d i x > l t h i c d x' > l , x ' ° > l , x " > l , x ' ° >
nen ( x ' - l ) ( x ' ° + x=' + x ' ° + x ' - 4) > d6 ta c6 y >
V d i 0 < X < t h i x ' < l , x ' ° < l , x " < l x " " < 1, nen ( x ' - l ) ( x ' ° + x " + x'° + x ' - 4) >
Do dd ta cd y > V a y m i n y = iTng v d i x =
D e 4.16: T i m gia t r i nho nhat cua >' = x^* - x ' +
74
T a c d : ); = ( x ' ° - x ' ) - ( x ' - - l ) + = x ^ ( x ' ^ - l ) - ( x ^ - l ) +
= x' \{x')' - ] - ( x ' - ) + = ( x ' - l ) ( x " ^ + x ' ' + x ' + x ' - ) +
V d i |x| > thi cd x " > x " > x ' > x ' >
=^ x ' - > va x'^ + x " + x ' + x ' - > nen y >
V d i |x| < thi cd x'^ < x'^ < x ' < x ' <
x ' - < va x " + x " + x ' + x ' - < nen y >
Do dd m i n y = k h i x =
Gi§i V i X + y = nen:
T a c d : M = 1 - 1
-x ^ /
( X + l ) ( x - X)iy ^\\y-1) (X + \){-y){y + l ) ( - x )
x ^ / x ^ /
(x + l)()^ + l ) _ x y + x + )^4-l _ ^ ^
x>' xy xry
• T a c d : ( x - y ) ^ > < i = » ( x - y ) ^ + x > ' > 4x3;
\
^ (-^ + yf > ^xy ma X + y = 1
Do d d : — > V a y A > + 2.4 = xy
Dau " = " xay ra X = y = ^ V a y gia t r i nho nhat cua M la
(39)Dat diTdc khi: x = y = —
Dd 4.18: Cho x, y > O v a x + y < l Tim gia tri nho nhat cua bleu thitc:
A =
+ xy -Axy
Gi^i
Ap dung bat dang thiJc Co si cho hai so' du'cfng a va b: a-\-b
(a + fef , 1 a b a + b
va
cib [a + b)
Ap d u n g ( l ) , (2), (3) ta c6:
(1)
(2)
(3)
A = 1
x' + r Ixy)
4
+ '^xy-\-4xy
5
+ - ^—2 T — ^ + 2,
x^+y^+2xy V
4xy h— xy
4
Vay : A > 1
4xy (x + y)
> + + = l l
1
Dau " = " xay ^ X = y - - Vay A dat gia tri nho nhat la 11
De 4.19: Tim gia tri Idn nha't ciia cac bieu thitc sau: 1) M = x y - x ' - y ' + x + y -
2) A^ = x y - x ^ - y ^ + x + 2y
76
3) P = — ( y z V x -1 + xzsjy -2+ xy.y/z - 3"
xyz ^ '
G i i i
1) Tacd M = - - ( x ^ 2xy + y ^ - x - y )
= - - [ ( x - y - l ) ' + y ' - y - l
= - - [ ( x - y - l ) ^ + ( y ' - y + ) -
( x - y - l f + ( y - r
=
-Xet ; c - y - l = y - =
X = >
y =
Vay khi X = 3, y = thi M dat gia tri Idn nhat la max M = 10 2) Tac6A^ = - ( x ^ - x y - x + y ^ - y )
\
+ | y ^ - y - l
x-^—X
\2
= - x - l - l
+ ( y ' - y + ) - 4
4
Xet 2 ^ x =
y = y - =
Vay khi X = y = thi N dat gia tri Idn nha't la max N = 3) Dieu kien xac dinh la: x > l , y > 2,z >
x y z
Dung ba't d i n g thiJc: a + b> 2yfab thi ta c6:
X = l+(X1)> 2y/x \ ^ <
-X
(1)
(40)y^2 + (y-l)>2^2(y-2)^^l^-^<~
I 2
• =:3 + ( z - ) > V ( z - ) ^ ^ ^ ^ < ^
(2) (3) Khi X = 2, y = 4, z = thi (1), (2), (3) trd dang thiJc Do d6 ta diTdc:
max P = — + 2 2^/2 ^ ' 4.20: Cho ba so' khong am x, y, z thoa man:
2x + y + 2z = (1) 3x + > ' - z - (2)
Hay tim gia tri Idn nhat va nho nha't cua M = 4x - 5y + 8z • Tir(l)tac6 :2x + y = - z
Tvlf(2)tac6 : 3x + 2y =:4 + 2z Suy rax = - z v a y = lOz-7
Vi x>Q^6-6z>0^z<\ 10
Talaico M = 4(6-6z)-5(10z-7) + 8z =69-66z
1 44 nen maxM = 69-66.—= — itng vdi mmz = —
min M = 69 - 66 = tfiig vdi max z =
78
CHlJOfNGV: P H I / O N G TRi]NfH
§1. PHaONGTRlNH BAG HAI - PHl/dNGTRlNH BAG BA
DINH LY VIET
I PHl/dNG TRINH BAC HAI ax^+bx + c = {a^O) Lap biet thitc: A = 6^ - 4ac
1) Neu A < thi phifcfng trinh v6 nghiem 2) Neu A = thi phiTdng trinh cd nghiem kdp:
X, =x^= —
la
3) Nd'u A > thi phifcfng trinh cd hai nghiem phan biet:
_ -^-VA _ - Z P + 7A
la
Trirdng hdp dac biet:
• Neu: a + + c = thi phu^dng trinh cd nghiem :
— - £ CI
• N6u: a - ^ + c = thi phu^dng trinh cd nghiem : x^-=^-{,x^^~-
a
Ghi chu: 1) Co the dung cong Ihitc nghiem thu gon de giai phUcfn I tilnh bac hai
2) N6u phu-dng trinh: ax^ + fejc + c = cd hai nghiem x^;x^ {x^ < j:2)thi: a > | : * ax^+bx + c<0<^ x^ <x<x^
X < X * ax'' +bx + c>0^
(41)a<0 * ax +bx + c<0<!^
* ax^ + bx + c>0'!F> X, <x<x^
JC < JCj
x> x^
I I D J N H L Y V I E T
1) Binh ly thuan: Neu phifdng trinh bac hai:
ax'^ +bx + c = ( a ^ 0) CO hai nghiem x^,x^ T h i :
* S =
x,+x^= a * P = x^.x^ = — a
2) Binh ly dao: Neu x + y = S va x.y = P thi x, y la nghiem cua phu-dng trinh: -SX + P =
I I I P H l T d N G T R I N H B A C B A ax' + bx^ +cx + d = 0 (a ^ 0) Cach giai: 1) Nham nghiem x^ = a
Dac biet: * Neu: a + b + c + d = thi phu'cfng trinh c6 nghiem
•.x^=l
* Neu: a - b + c - d = thi phu'cfng trinh cd nghiem : X o = —
2) Chia ax' + bx' +cx + d cho x — adi du'a phrfdng trinh ye dang:
(x-a)(Ax'' +Bx + C) = • (*) x-a^O
Ax^ + Bx + C = 0'
T a c d : ( * ) ^
iS.l: Cho phu'cfng trinh: x^ -2(m - 2)x - m - =
1) ChiJng minh r^ng phu'cfng trinh cd hai nghiem phan biet v d i moi m
2) Goi x^,X2 la nghiem cua phu'cfng trinh T i m gia tri m sac cho: xf +xl =
80
1) T a c d : A ' = Z?''-ac = ( m - ) ^ - ( - m - ) = m ^ - m + - ( m - l ) ' + =!^A'>OVm
Vay phufdng trinh da cho ludn cd hai nghiem phan bi6t v d i moi m
2) Theo dinh ly Viet ta cd:
x^+X2= = 2(m - 2)
a
X , x^ = — — —2m —
a
Mat khac theo gia thiet:
xf+xl=lS<:^(x^+x^f-2x^x^=lS
^ [2(m - 2)f - ( - m - 5) = 18 <^ m ' - 16m + + 4m + = 18 <^ m ' - m + = <!=J> m ^ - m + =
m = 1V m = (pt cd dang a + b + c = 0) Vay m = hoac m = thi xf + A:^ = 18
Bi 5.2: Cho ph^dng trinh bac hai: x^ - 2mjc + 2m - = (1)
1) Dinh m de phu'cfng trinh (1) cd nghiem kdp
2) V d i gia tri nao cua m thi phu'cfng trinh (1) cd hai nghiem cung dau K h i dd hai nghiem mang dSu gi ?
Gi§i
1) T a c d : A ' = ^ ' ' - a c = m ' - ( m - l ) = m ' - m + l = ( m - l ) ' Phu'cfng trinh (1) cd nghiem kep <i=^ A ' =
< ^ ( m - l f = < ^ m = l 2) PhiTdng trinh (1) cd dang dac biet:
a + b + c = l - m + m - l = nen cd nghiem: =
(42)— — = 2m —
a
• Phifcfng tnnh (1) CO hai nghiem cung dau
X j j T j > <^ (2m -1) >
2m > <^ m > -2 y
• Khi do: = Xj + jc^ = = 2m > nen J:, ^"^^8 difdng Tdm lai: Khi > ~ '^hi phu'dng trinh (1) cd hai nghiem cung dau va dd hai nghiem ciing du'dng
Ghi chu; Cd the nhan xet: x^ va x^ cung dau va = > n6n x^>0
5.3: Cho phufdng trinh cd an x: - 2(m - - - m = 1) Chitng to phufdng trinh cd nghiem so* vdi moi m
2) Tim m cho nghiem so' x^,x^ ciia phtfdng trinh thda dieu kien: xl + x] >10
1) Tacd: A ' = (m-1)^+3 + m m ^ - m + 1+ + m = m-m-\-A = m-m^
4 1
m 2
' 15 + — > Vm 4
Vay phrfdng trinh cd hai nghiem phan biet x^,x^ vdi moi m x^ -\-x^ =2(m-l)
x^x^ = —3 — m 2) Theo he thiJc Viet ta cd:
Matkhacdo: xf +xl > 10 <^ (JC, + ^2)^ -2x^x^ >10 82
<^[2(m-l)f-2(-3-m)>10<^4(m'-2m + l) + + 2m>10 <^4m^ -8m + + + 2m >10<^4m^-6m>0
m < 4^
m "
ay khi: m < hoac m > - thi xf + x\ 10
De 5.4: Giai phufdng trinh : - (2m + 1)J: + m^ + m - = 1) Dinh m de phu'dng trinh cd hai nghiem deu Sm
2) Dinh m de phufdng tiinh cd hai nghiem x^,x^ thda x^,-xl = 5C
Giii
Tacd: A = (2m + l ) ' - ( m ' + m - )
= 4m^+4m + l^-4m^-4m + 24 = 25 Phu'dng trinh cd hai nghiem phan biet:
2m+ 1-5 ' 2m+ + ,
X, = =:m-2;A:, = = m +
' '
1) Phu'dng trinh cd hai nghiem deu am
4 ^
•ill
x,<0 m - < m <2 •^2 < o m + < m < -
2) Tacd: = 50<^ ( m - ) ' - ( m + 3y = 50 m' - 6m' +12m - - m' - 9m' - 27m - 27 = 50
-15m?-15m-35 = 50 <!=^ 3m'+3m + =10
(43)3m^+3m + = 10 m ' + m + = -
3/n^ + m - = (1) m ' + m + 17 = (2) • Giai (1) : 3m' + m - = < = > m ' + m - l =
A = l + =
- I + VS - l - ^ / ' '
^2
• Giai.(2) : m / + m + 17 = A = - = - < Phifdng trinh v6 nghiem
- l + ^/5 , „ -\-yl5 , , ^_ Vay m = ^ hoac m = ^ thi phifdng trmh da
cho C O hai nghiem x^,x^ th6a man x^^ -x] = 50
5.5: Cho phifdng trinh: x^ - 2(m + 1)A: + m ' - 4m + = (c6 an so l a x ) '
1) Dinh m de phifdng trinh c6 nghiem
2) Goi x^,x^ la hai nghiem neu c6 cua phifdng trinh Tinh
xf + ^ 2 theo m
1) T a c o : A ' = ' ' - a c = (m + l ) ' - ( m ' - m + 5) = + 2m + - m ' + 4m - = 6m -
2 Phifdng trinh da cho cd nghiem ^ A ' > < ^ m - > < ! ^ m > -
Vky m > J thi phifdng trinh da cho cd nghiem
2) Theo he thiJc Viet ta cd: S = x^+x^= 2(m + 1) = 2m +
P = x^x^ = m ' —4m +
84
Do do: xf +xl =(x^ + x^f -Ix^x^^ (2m + 2f-2{m^-4m+ 5) I = m ' + m + - m ' + 8m - = m ' + m -
Vay: A : , ' + m ' + m -
Bi 5.6: Cho phifdng trinh: x ' - 2(m + l)x + m ' + 2m - = (1) 1) Chitng minh phifdng trinh (1) lu6n cd hai nghiem phan biet
vdi moi m T i m hai nghiem dd 2) T i m m de phifdng trinh (1)
a) Cd hai nghiem trai dau b) Cd hai nghiem deii difdng c) Cd hai nghiem deii am
G i ^ i
1) T a c d : A ' = Z?''-af = (m + l ) ' - ( m ' + m - )
= m ' + m + l - m ' - m + =
V i A ' = > OVm n^n phifdng trinh da cho lu6n cd hai nghiem phan biet
K h i d d : X =
^ = •
= m + l - = m - l
= m + l + = m +
) a) R6 rang: x^ < x^ nen phifdng trinh (1) cd hai nghiem trai dau
m - l <
^ x^ <0 < x^ 4^
m <
x,<0
X , > m + > - < m <
m > -
(44)X , > X , >
m - > m >
m > - m >
[m + >
V a y k h i : m > thi phifdng trinh (1) c6 hai nghiem deu difcfng c) PhiTcfng trinh (1) c6 hai nghiem deu am
x,<0 x^<0
m - K O m + <
m <
m < - <^ m < - V a y k h i : m < - thi phu-dng trinh (1) c6 hai nghiem deu am
5.7: Cho phtfdng trinh : - (2m - 3)x + - 3m =
1) ChiJng minh rang phu-dng trinh luon luon c6 hai nghiem k h i m thay d d i
2) D i n h m de phiTdng tiinh c6 hai nghiem x^,X2 thoa:
\<x^<x^<6
GiSi
1) Ta c6 : A = (2m - ) ' - ( m ' - 3m)
= m ' - m + - m ^ + m =
A = > Vm V a y phtfdng trinh l u o n l u o n c6 hai nghiem phan biet k h i m thay d d i
2) V A = H a i nghiem cua phu'dng trinh la: m - -
= m — 3;x. =
2 m - + = m
2 '
R6 rang < x^: N e n : \ x, < x^ <6 ^ Km-3 <m <6 K m -
m<6 <^ < m <
4 < m m <
86
5.8: Cho phu'dng trinh: - (2m - 5)x + m^ - 5m =
1) Chi?ng minh rang phu'dng trinh luon c6 hai nghiem phan biet k h i m thay d d i
2) D i n h m de phufdng trinh c6 hai nghiem thoa: a) D e u am
b) D e u diMng c) T r a i dau d) < < < ^
e) x^<2<x^<6
Gi§i
1) T a c o : A - Z ? ' - a c = ( m - f - ( m ' - m ) = 4m^ - 20m + 25 - 4m^ + 20m = 25
V i A > OVm nen phu'dng trinh lu6n c6 hai nghidm phan biet: -b-sfA m - -
K h i 66 : X, =•
la = m -
x^ = -b + y[A m - + 2a 2) a) Phufdng trinh cd hai nghiem deu a m
b) PhiTdng trinh c6 hai nghiem deu dtfdng
• = m
X < m - < < m <
m <
m < <^ m <
x,>Q m-5>0 x^>0 m >
m >
m > <^ m > | c ) R6 rang: x^ < x^ nen: Phu'dng trinh c6 hai nghiem trdi d ^
m <
X. < m - <
< ^
x^>0 m > m >
(45)X, <2 m - <
6 < ^2 6<m
m <1
6 < m < < m
e) Phifdng trinh c6 hai nghiem X j, thoa: x^ <2<x^ < m - < < m < m - <
2 < m <
m < t
2 < m < ^ < / n < De 5.9: Cho phiTdng trinh: x^ - 3(m + l ) x + m ' - = (c6 an x )
1) T i m m de phifdng trinh c6 hai nghiem phan biet deu am 2) G o i x^,X2 la hai nghiem cua phtfdng trinh T i m m de c6:
< x^ X2
Gi^i
1) T a c : A = & ' - a c = 9(m + l ) ' - ( m ' - ) = m ' + m + - m ' +
= m ' + m + 81 = (m + ) '
V i A = (m + ) ' > Vm nen phu'dng trinh luon c6 hai nghiem: 3(m + l ) - ( m + 9)
•^1 = • = m —
^2 = •
3(m + l ) + (m + 9)
= 2m +
Phu'dng trinh c6 hai nghiem phan biet deu am A >
X i < <^
X^KO
(m + ) ' >
m - < ^ 2m + <
m < < ^ - ^ m < - m < -
D6 5.10: Cho phu'dng trinh: x^ - 2{m + l ) x + m ' - 4m + = 1) D i n h m de phu'dng trinh c6 nghiem
2) Dinh m de phu'dng trinh c6 hai nghiem phan biet deu dtfdng
88
Giii
1) T a c o : A ' - ( m + l ) ' - ( m ' - m + 5) = m -
Phu'dng trinh da cho c6 nghiem • ^ A ' > < ^ m - > < ^ m > — ~ "
2) V d i dieu k i e n > - phu'dng trinh da cho cd hai nghiem: X, = m + - V m - ; = + + V m -
Phu'dng trinh da cho c6 hai nghiem phan biet deu dufdng
A ' > x,>0 ^2 >
2 m >
-3
1 ^
6m - >
m + - V6m - > m + + V6m - >
- ( m + ) < V m - < (m + )
m > -3
m + l > ^ \/6m - <l m + 11
9
m > -3
6 m - < m ' + m + l
2 m > -3 m > -
0 < m - < m ' + 2m + l
2 m >
-3
m^ - 4m + >
m >
-3 m > - 2 ( m - ) ' + l >
V a y k h i m > - phu'dng trinh da cho cd hai nghiem phan biet deu du'dng
^^Qishu^ I) Cho A>0.T^ic6: X <A^-A<X < A ,
(46)2) Bai toan tren c6 the giai nhiT sau:
• Phifdng trinh da cho c6 hai nghiem phan biet deu difcfng A >
jc, > A , >
A > A, x^ >
A > P > S>0 ^
6m - > - 4m + > 2(m + 1) >
2 m >
-3 m G <^ m > —
3 m > -
Vay m > -J phi/dng trinh da cho c6 hai nghiem phan biet deu dtfOng
3) Bang each chiJng minh tufcfng tuf, ta cd ke't qua:
Cho phu'Ong trinh bac h&i: ax^ + bx + c =^ (a ^ 0)
• Phu'Ong trinh c6 hai nghiem phan biet trai dau <^P<0 'A>0
PhtTcJng trinh c6 hai nghiem phan biet deu du'cfng <^
PhUdng trinh c6 hai nghiem phan biet deu a m <^
P > > A > P > S<0
Bi 5.11: Cho phu'Ong trinh: mx^ - 2(m + 2)A + m = (c6 an x) D i n h m d e :
1) Phu'dng trinh cd nghiem
2) Phu-dng trinh cd hai nghiem trai daii 90
3) Phu'dng trinh cd hai nghiem phan biet deu am 4) Phu'dng trinh cd hai nghiem phan biet deu diTdng
GiSi
1) X e t hai trrfdng hdp:
• m = 0: phu'dng trinh ^ O.x^ - 4^ + =
<^ A — : phu'dng trinh cd nghiem • mr;^0: phu'dng trinh cd nghiem<^ A ' >
<»(m + ) ^ - m m >
<^4m + > - ^ m > - l (m^O) Tit hai tru'dng hdp tren, ta cd:
Phufdng trinh cd nghiem • o m > -
2) • Phu'dng trinh cd hai nghiem trai dau <^ P < <^ < (v6 nghi6m)
3)
m
Vay khong cd gia tri m nao thoa yen cau bai toan • Phu'dng trinh cd hai nghiem phan biet deu a m
A > P > ^ S<0
4m + > ^ > m m +
< m
wi > -
m + 2<0 vci m>0 m + 2>0 va m<0 m > - l
m > - va m > < ^ m > —2 va m <
m > -
me0 < ^ - l < m < - < m <
Vay khi: - < m < phu'dng trinh da cho cd hai nghiem phan biet deu am
4) • Phu'dng trinh cd hai nghiem phan biet deu du'dng
(47)A > P>0 <^ S>0
4m + >
^ > m m +
>
m>-l
m + 2>0 va m>0 m + 2<0 va m<0 m
m>-\
m>-2 va m>0<^ m<-2 va m<0
m>-\
m > 0 <^m>0 m<-2
Vay k h i : m > phifdng trinh da cho cd hai nghiem phan biet deu diTcfng
Bi 5.12: Cho phiTdng trinh: (m +1)^ - 2(m + 2)^: + m - = (cd an so l a x )
1) Dinh m d^ phiTcfng trinh cd nghiem
2) Dinh m de phuTdng trinh cd hai nghiem x^,x^ thda: (4x, +1)(4X2 +1) = 18
Giii 1) * V d i m + l = ^ m = - l
PhiTdng trinh ird : -2x - = 0>^2;c = -4<i^;c = - Vay m = - phiTdng trinh cd nghiem x = -
• V d i m + <^ m ^ - A ' = (m + ) ' - ( m + l ) ( m - )
= m^ + 4m + - w ^ + m - m + = 6m + Phirdng trinh cd nghiem 4=^ A ' >
^6m + l>0^6m>-l ^m> - ^ ( m ^ - l ) K^'t luan: PhiTcfng trinh cd nghiem <=> w > - -
6
92
2) D i e u k i e n O T > —- m ^ - l
Theo dinhly Viet ta cd:
5 = + = 2(m + 2)
P = JC, V =
m + m -
Do dd: (4^1 + l)(4;c, + 1) = 18 •'-> \6x,x^ + 4;Cj + 4x, + = 18
«!=J> 16 A:, A:^ + 4(x, + J C j) - =
1 ( m - ) , 8(m + 2)
17 =
m + m + • 16(m - 3) + 8(m + 2) - ( m +1) =
<^ 16m - 48 + 8m +16 - 17m - = 7m - 49 = 7m = 49 <i=^ m =
m = thoa dieu kien
m —1
m > - Z ~
Vay m = thi phiidng trinh cd hai nghiem x^,x^ thoa : (4x, +1)(4X2 +1) = 18
1)6 5.13: Cho phu-dng trinh: x^ - 2(m - l)x + 2m - = ^ (cd an so' la x)
p 1) Chitng minh rang phiTdng trinh cd hai nghiem phan biet
^ I l ^ 2) Goi x^,x^ la hai nghiem cua phifdng trinh Tim gia tri nho nhatcua y = x^ + x^
Giii
1) Tacd : A ' = ( m - ) ' - ( m - ) = m ' - m + l - m + = m ^ - m + + l = ( m - ) ^ + l >
Vay phu-cfng trinh cd hai nghiem phan biet vdi moi m
(48)2) Theo dinh ly Viet ta c6: x ^ + x ^ = 2{m - 1) = 2m -
X j = 2m —
y + xl = (jc, + ;c, )^ - 2XjX, = (2m - 2)' - 2(2m - 4) = m ^ - m + - m + = 4m^-12m + 12
= ( m ) ' - 2.2m.3 + ' + = (2m - / + >
Dau " = " xay ^ 2m - = 0 4=^ 2m - ^ m = - 2
Vay y = dat difdc m = —
5.14: Cho phufdng trinh - (2m + 3)x + m - = (cd an la x) 1) ChiJng to rang phifOng trinh luon c6 nghiem
2) Goi la cac nghiem cua phu'dng trinh tren Tim m de
-C[ X2 dat gia tri nho nha't Tinh gia tri nho nha't ay
Gi^i
1) A = (2m + 3)' - ( m - ) = 4m- +12m + - m + 12 = m ' + 8m + + 17 = ( m + ) ' + >
Do phu'dng trinh c6 hai nghiem phan biet vdi moi m \x, + x^ =2m +
2) Theo dinh ly Viet ta cd •
[x^X2 = m —
Mat khac ta cd: (x^ - x^ f = (;c, -\- x^f -Ax^ x^
= (2m + 3)' - 4(m - 3) = (2m + 2)" +17 > 17
Do dd \x, - X \ yl(2m + 2f +17 > Vl7
Dau "=" xay <^ 2m + = <s=^ 2m = - <^ m = -
Vay x^-x^ = Vl7 dat du'dc m = -
5.15: Cho phufdng trinh x^ - 2mx + 2m - = vdi m la tham s6'
94
1) ChiJng minh phu'dng trinh luon c6 nghiem
2) Tinh bieu thitc y=— 5 theo m (vdi x,,x, x; +xl+2(\ x^x2) " ^ la nghiem cua phu'dng trinh)
3) Tim gia tri Idn nhat va nho nhat cila y
1) Tacd A ' = m ' - ( m - l ) = ( m - l ) ' > vdi moi m nen phifdng trinh luon cd nghiem
2) T a c d : S = x^ + x^ =2m,P = x^x^ =2m-\ D o d d : y = ^ ^ • ^ ^ + ^ ^^x^x^ + 3_
x^ +xl+2x,X2+2 (x^+X2f+2
_ 2(2m - ) + _ 4m + m ' + ~ m , ' + m +
3) Tu^ y = suy ra: m ' +
4 y m ' + y = 4m + l < ^ > ' m ' - m + y - l = (*) Ta dijng dieu kien cd nghiem ci\ phu'dng trinh de tim y va max y
• N e u y ^ O : (*) <i=^-4m - = <^ m = - - ( p t (*) cd
nghiem)
• Neu y^0:{*)c6 nghiem
A ' = - y ( y - l ) > ^ y ' - y - l <
^ - | < y < i ( y - ^ )
Tiir hai tru'dng hdp tren ta cd: pt (*) cd nghiem
V a y m i n > ' = - — va max>' = l
(49)5.16: Cho phiTdng trinh : - 2mx r = •
m
1) T i m m de phifcfng trinh CO nghiem
2) Dat A = x," + xl vdi x^,x^ la nghiem cua phiTdng trinh neu tren T i m gia tri nho nhat ciia A
Gidi
1) Ta cd A ' = + — > vdi moi m ^ n^n phifdng trinh cd
m
nghiem m
4 2) Theo dinh ly Viet, ta cd: x^+x^= 2m,x^x^ = ^
J m
A^ixlf+{xlf={x]^xlf-2xlxl
{x, + x,f-Ax,x, -l{x,xS = f
2 2
- ] 2
4 m ' + • + ^ = 16
m m + •
32 m = 16 4 I 32
m ) m m ^ + - ^ + m
Mat khac theo BDT Cosi ta cd:
m \ m
Dau " = " xay = ^ = ^ m = ± ^ m
Va luc A dat gia tri nhd nhat la: A = 16(2V6 + 4) = 32(V6 + 2)
D(! 5.17: Cho phufdng trinh : x ' - 2(m + ) + 2m = (1) 1) ChiJng to rang phu-dng trinh (1) luon lu6n cd hai nghiem phan
biet vdi moi gia tri cua m (
96
2) Goi x^,x^ la hai nghiem ciia phUdng trinh (1) Chifng to rang gia tri ciia bieu thitc x,-]r x^-x^.x^ kh6ng phu thudc gid tri ci5a m
1) Ta cd: A ' = ^•' - = [-(m + l)f - 2m = m ' + m + l - m = m H l V i m ' > n e n m ' + l > O S u y r a A ' >
Vay phiTdng trinh : x ' - ( m + + 2m = luon luon cd hai nghiem phan biet vdi moi gid tri cua m
2) Phu-dng trinh : J : ^ - ( m + l ) j : + 2m = luon luon cd hai nghiem phan biet nen theo dinh ly Viet, ta cd:
x^=2{m-\-\) va x^.x^—2m
T a c d X, +x^ -x^.x^ = ( m + l ) - m = 2m + - m =
Do dd, bieu thiJc x^ x^ -x^.x^ khong phu thuoc vao gia tri ciia m
5.18: Cho phtfdng trinh: (m + \)x^ - 2{m-\)x + m-2 = Q 1) Xac dinh m de phufdng trinh cd hai nghiem phan biet
2) Xac dinh m de phtfdng trinh cd mot nghiem bang va tinh nghiem
3) Xac dinh m de ph^dng trinh co hai nghigm phSn bid! x^,x^ thoa man he thitc
x^ x^
Gi§i
# Cho phutfng trinh (m + \)x'^ - 2(m - + m - =
1) Phu-dng trinh : ( m + - ( m - 1)A:+ m - = Ocd hai nghiem phan biet va chi
(50)m + ^ A ' >
m ^ —I
- ( m - l ) f -(OT + l ) ( m - ) >
OT ^ —
m - m + l - m ^ + w - m + > - m > -
m ^ —1 m <
Vay k h i m < va m ^ - thi phiTcfng trinh trSn c6 hai nghiemphan biet
2) Phtfdng trinh (m +1)^:^ -l{m-X)x + m-l = 0c6 mot nghiem bkng k h i chi k h i
(^4-1)22 - ( m - l ) + m - = ^ m + - m + + m - =
Vay k h i m = - thl phifdng trinh (m + \)x^ - 2(m - l)x + m - = c6 mot nghiem bkng
Khi m = - phiTdng trinh (m + - 2(m - ^ m - = c6 hai nghiem phan biet x^,x^ nen theo dinh Viet ta c6:
2 ( m - l ) ( - - l ) _ - + m +
14 14
Vay nghiem lai cua phiTdng trinh tren \k:x^= —
3) Ta x6t trirdng hdp phifdng trinh c6 hai nghiem phan biet x^,x^ tiJclk: m < va m ^ -
2 ( m - I ) m - Theo dinh ly Viet ta c6 x.+x^= va x, x^ =
m + /n +
Phifdng trinh (m + l ) ; c ^ - ( m - l ) x + m - = 0c6 hai nghiem x^,x^ th6a man he thiJc
98
^ ( m - l ) _ m-2 ^ ^ ( m - l ) _ m + m + m - ~
m vt2
OT = —6 m:;^2
8 m - = m - 4=^ /n = -
Do m = - thi phUdng trinh
(m + l ) j r ' - ( m - l);c + m - = 0c6 hai nghiem phan biet x^,x^ thoa man he thitc — + — =: —
D d : Cho phifdng trinh : - ( m - l)x + m - =
1) Chitng minh rang, vdi moi gia tri ciia m phiTdng trinh luon c6 hai nghiem phan biet
2) Goi x^,x^ la hai nghiem ci5a phUdng trinh da cho T i m he thiJc lien he giffa x^,x^ doc lap do'i vdi m
Gi^i
1) Ta CO A ' = b''~ ac
- ( m - l ) f - ( m - ) = m ' - m + l - m + = m^ - m + = m
9
Do dd, phu-dng trinh x^ - ( m - l ) x + m - = luon luon cd hai nghiem phan biet v d i moi gia tri ciia m
2) PhUdng trinh - ( m - l ) x + m - = 01u6n luon cd hai nghiem phan biet x^,x^ nen theo dinh ly Viet ta c6
x^+x^=^2{m-\) x^x^ = m —
(1) (2)
(51)Tu" ( ) t a c : m = X i X + T h a y v a o ( l ) :
x,+x^^ lx,x^ + =^ lx,x^ -x,-x^+A = Q
He thiJc lien he giffa x,,x^ doc lap doi vdi m la :
l x ^ x ^ - x ^ - x ^ ^ A = Q
5.20: PhiTcfng trinh : 2J:^ - 6x + m =
1) Vdi gia tri nao cua m thi phu-dng trinh c6 hai nghiem deu du'dng?
2) Vdi gia tri nio cua m thi phtfdng trinh c6 hai nghiem x^,x^ sao cho :
+ — =
x^ X, •
GiSi
1) Phu-dng trinh 2x^ -6x^m = Q c6 nghiem
<^ A ' > < ^ ( - ) ' - m > < ^ - m > 9
2
Khi phu-dng trinh tr^n cd hai nghiem x, va x,, theo dinh ly Viet ta c6 : X, = va x.Xj = - J
Phifdng trinh 2x^ - 6x + m = c6 hai nghiem dufdng khi: [ A ' >
5 = X, + Xj > 0 <s=^
P = x,.X2 >
~2
3 > 4=> ^ > 2
9
m < - ^ ^ ~ < ^ < m < -
m > ^ Do dd < m < I thi phtfdng trinh 2x^ - 6x + m = c6 hai
nghiem dufdng 100
2) Khi w < - ~
Phu-dng trinh cd hai nghiem Xj Xj thda ^ + ^ =
Xj Xj
x f + x ^ ^ (x, + X ) ' - x , X
=
Xj x^ x,.x,
( ) ' - m ,
^ — = 4=^ m 18 — 2m = 3m
m :;^0 18
l ^ ^ m = — ( t h a d k m < - )
m = — 18
Do dd '^ = — ^hi phufdng trinh 2x^ - 6x + m = cd hai nghiemx, x, thoa :
^ + ^ =
Xj Xj
DC' 5.21: Cho phu-dng trinh bac hai do'i vdi x :
(m + l ) x ' - ( m - l ) x + m - = ( m ^ - ) (1) 1) Chitng minh r^ng phifdng trinh (1) luon luon cd hai nghiem
phan biet vdi moi gia tri ciia m
2) Goi Xi,X2 la nghiem cua (1), tim m de x^x^ > va Xj = 2x2
1) Tacd A' = [ - ( m - l ) f - ( m + l)(m-3) = m^ - m + l-(m.^ - m + m - ) = - 2m +1 - m^ + 2m + = >
(52)Vay phu-cfng trinh (m + l)x^ -2(m -l)x + m-3 = luon luon c6 hai nghiem phan bi6t vdi moi g i i tri cua m ^ -
2) Do phu-dng trinh (m + \)x^ - 2(m - l)x + m - = luon luon c6 hai nghiem phan bidt v d i moi gia tri cila m ^ - nen theo dinh 1^ Viet ta cd :
2 ( m - l ) m -
x,+x.= — va X. X, = ' ' m + ' ' m +
Phu-cfng trinh (1) cd hai nghiem x^ ,x^ thoa man = 2x^
2x^ +x^ = 2 ( m - l )
m +
2x^.x^ = m -
m +
•^2 = 2 ( m - l ) 3(m + l ) m —3 2(m + l )
(1) (2) Binh phtfdng (1) va so sanh vdi (2) ta diTdc:
4 ( m - ) ^ ^ _ m - _ ^ , _ m + ) = 9(m^ - m + m - ) 9(m + l ) ' 2(m + l )
8 m ' - 16m + = m ' - m - 27
K h i m = ta cd: x,.x^ =
- m - = m - - I
mj = = —5
K h i m = - ta cd: x^.x^ =
' m + + m - - -
= - > = > m + - +
Do dd m = hoac m = - thi phu-dng trinh (1) cd hai nghiem thoa man x^ = 2x^ vk x^x^ >
Bi 2 : Cho phu-dng trinh x^ -ax + a + \ ChiJng minh r^ng n6'u •.a + b>2 thi it nhat mot hai phu-dng trinh sau day cd nghidm : x^ + 2ax + b = 0,x^ + 2bx + a =
102
Giii
Phu-dng trinh : x^ + 2ax + b = c6 biet so' A\=a^ -b Phu-dng trinh -.x^ + 2bx + a = cd biet so' A\=b^ -a Suyra A' l + A ' j = a ' + Z J ' - ( a + Z?)
= ( a ' - a + l) + ( ^ ' - Z j + l) + (a + ^ - )
= (a-lf +(b-lf +(a + b-2)>0
( V i (a-\f > , ( ^ - ) ' > v a a + ^ > (gt) nen a + b-2>0) =^ ft nhat mot hai biet so' A ' j , A cd mot so' khong am
That vay ne'u ca hai am nghla la A ' , < va A'^ <
A'l + A'^ < (vd ly)
Vay it nhat mot hai phu-dng trinh da cho cd nghiem
5.23: Chii-ng minh hai phufdng trinh sau cd it nhat mdt phu-dng trinh cd nghiem:
ax^ +bx + c^O (1)
ax^+cx + b-c-a = 0 (2) (vdi a * 0,^ e/?,c G/?)
Giii
(1) c d b i e t s o l a A , = Z ? ' - a c
(2) cd biet so la A^ = c' - 4a(b -^c -a) = c^ - 4ab + a ' + 4ac Tacd A , + A , =Z>' + c ' - a i + a '
= (b^-4ab + 4a^) + c^ = ( b - a f >
V i A j + A j > nen suy A , > hay A j > Vay (1) cd nghiem hay (2) cd nghiem
I>6' : T i m k de hai phu-dng trinh sau cd nghiem chung:
x^+kx + \=0 (1)
x^+x + k = 0 (2)
(53)-Gidi
Goi la ng^^iem so' chung thi ta c6:
xl+kx,+l = 0 (1')
xl+x,+k = 0 (2') Tru" ve'(1') cho (2') thi c6: (k-l)x,+ l-k =
^(k-l){x, l)=^0
<^ k — I hay = ^
• V d i k = thi (2) trd + A; + = Day Ik phuTdng trinh v6 nghiem
• V d i = 1: the vao (2') ta c6 k = - Liic d6, ta cd:
(1) ^ x^ -2x + l = 0^ x = \
(2) ^ x''.+x-2 = 0^ x = l,x^-2
Do hai phifdng trinh cd nghiem chung la x =
5.25: Cho a, b, c la dai ba canh cua mot tam giac QnJng minh phu'dng tiinh sau la v6 nghiem: a^x^ + (a^ + — c^)x + =
Gi§i
Taco A = (a'+ b'-cy-4a'b'= (a'+b'-c')^-{2abf = (a^+b^~c^-2db)ia^+b^-c^+2ab)
= \a-bf-c^].[{a + b)''-c^
= {a-b-c)ia-b + c)(a + b + c){a + b-c)
V i a<b + c^a-b-c<0
b<:a + c^a + c-b>G c<a+b^a+b-c>0
va a + b + c>0 nen A < va do phu'dng trinh tren v6 nghiem-De 5.26: T i m m de 2x^ - 3x + 2m = c6 mot nghiem khac va gap ba Ian mot nghiem cua 2x^ — x + 2m —
104
Gi^i
X6t 2x^-x + 2m = 0 (1)
2 x ' - j c + 2m = (2) Goi XQ la nghiem cua (1) cho 3XQ la nghiem cua (2) thi ta c6:
2xl-x,+2m = 0 (1')
2 ( 3; C o/ - ( X o ) + 2m = (2')
Trir ve (2') cho (1') thi: 16x1 -^x^ = ^ x^ = 0,Xo
V i dieu kien cua de bai \& x^ ^0 nen ta chi xet -^o = ^
1 1
The = - vao ( l ' ) t a du-dc: + 2m = ^ m =
Liic d6 (I) ^2x^-x = 0^ x = 0,x = ^
(2) < ^ x ^ - x = ^ x = , x - |
nghiSm = ^ cua (2) la gap ba Ian nghiem = ^ cua (1)
Vay m = la gia tri can tim
b e 5.27: T i n a va b de hai phu'dng trinh sau la tu-dng du'dng
x''-(4a + b)x-6a = 0 (1) •
x^ -(2a + 3b)x-6 = 0 (2)
Giii
Neu (1) va (2) la tu-dng diTdng thi (1) va (2) c6 nghiem trung
Do 5 , = , 4a + ^ = 2a + 3Zj - a = - 4^ a = \,h = \
(54)L i l c ay, ta c6: (1) - x - = <^ ;c =
V a y a = b = la dap so
-\,x = - \ , x ^
D e : Cho hai phiTdng trinh :
x^ -{2m-3)x + =
+ X + m - = (x la an, m la tham so') T i m m de hai phifcfng trinh da cho c6 diing mot n g h i f m chung
Gi§i
Gia su" XQ la nghiem chung cua hai phifdng trinh
Ta CO
xl - (2m - 3)Xo + = xl-(2m-3)x,+6 = 0 (1)
m = —2XQ — +
2x1 + ^ 0 +m — — Thay m t i r ( ) va (1), ta c6:
JCQ^ - [2(-2;Co^ - + 5) - 3] + =
<^ xl - 4^ 0 + 2^ 0 - 7^ 0 + =
^4x1+3x1-1x^+6 =
^ + 8^ 0 - 5^ - I O X Q + 3^ 0 + =
<^ 4xl(x, + 2) - {x, +2) + 3{x, + 2) =
^ix,+2)(4xl-5x^+3) = j C o+ =
4x1-5x^+3 =
(2)
44>
X =—2 XQ — L
x^e0 ( v i A = - = - < ) V d i : x^ = -2 thay vao (2), ta dufdc:
m = - ( - f - ( - ) + = - l
vay hai phtfcfng trinh c6 mot nghiem chung la -2 k h i m = -
106
^^fs^- G i a i cac phu-dng trinh bac ba sau :
\) +2x^-lx + A = 2)x'-2x^-x + = Gi§i
1) x' +2x^-lx + = 0 (1)
De thay phu'dng trinh c6 dang: a + b + c + d = nen c6 mot nghiem: x =
Chia: x^ + 2x^ - lx + 4 cho x - 1, ta du-cfc:
{\)^{x-\){x'' + 3A: - ) =
U = l
44-x - \
; C 2+ 3J C- =
x = \ x = \y x = -4
x = - 2) x' -2x^ - x + = 0 (2)
De thay phu'dng trinh c6 dang: a - b + c - d = nen c6 nghiem: x = Chia: x^ - 2x'^ - x + = 0 cho x + 1, ta drfdc:
( ) < ^ (A : + ) (X '- x + 2) =
U = - i ,
-
4^
x + \
x^-3x + =
x = \ x =
5 : Cho so x^ = ^ + 4^5 + ^ / 9- W
1) ChiJng to XQ la nghiem ciia phu'dng trinh x^ - 3x-\% =
2) T i n h
1) T a c d :
(55)= + ^ / + ^ + ^ - 5 (V9 + 4V5 + ^ - ? ' + - - ( ^ + 475 + V - 5] -
= 3781-80(^9 + 5+ ^ 9- ] -3 [^9 + 475 + ^ -4 ] =
Vay = V9 + 475 + ^ / - 5 la nghiem cua phiTdng trinh:
A; ' - 3A; - -
2) X6t phircfng trinh: x ' - x - = (*) Bkng each nham nghiem ta thay phu'dng trinh (*) c6 mot nghiem x =
Chia: A:^ - 3;c -18 cho x - Ta dtfcJc:
(*)^(x-3)(;c^+3x + 6) = \x =
+3x + = 0(VN) -ip^ x =
Vi phu'dng trinh (*) cd mot nghiem nha't x = nen: x^ = D^5.31: Chophu'dngtrinh: ( ^ - 2) ( x ' - A : ) - ( x- ) ( A : - m ) = (1)
1) Giai phu'dng trinh (1) m =
2) Vdi gia tri nao cua m, phu'dng trinh (1) cd ba nghiem phan biet?
= Ta cd :
(x -2)ix^ -x)-(x-2)(2vC-m) = 0^{x-2)[(x' -x)-(Ix-m)
4=> (;c - 2)(x^ - ;c - 2JC + m) = ^{x-2){x'-3x + m) = Q 1) Khi m = phu'dng trinh (1) trdjhanh: ^ ^ , , 108
{x-Dix" -3x^\) = 0<^ x;c2_3x- 2 = + l =
^2_3;c + l = d x,=2 ^2 =
X3 =
3 + 75 2 - Vay m = thi phu'dng trinh (1) cd ba nghiSm phan biet
+ 75 -
X = 2;x = va x =
^x-2 =
x^ - 3x + m = > x =
x^-3x + m = 2) Tacd: ( A:-2)(x'-3x + m) = ^
44>
Tir dd suy phifdng trinh (1) cd ba nghiem phan biet va chi phifdng trinh ;c' - 3x + m = cd hai nghiem phan biet khac
A >
2'-3.2 + m ^ - m > -
9 - 4m > - + m ^ O m ^
9 m < -4 m ^
Vay m < - va m ^ thi phtfdng trinh (1) cd ba nghiem phan biet
(56)§ 2. PHUdNG TRINH Q U I VE B A C HAI
I Phifong trinh trung phiforng: ax" + bx^ + c = (a ^ 0) Cach giai: Dat t = x^ (dieu kien: ; > 0) phifdng trinh trd thanh:
at^ + bt + c =
• Giai va kiem tra dieu kien < >
II PhiTorng trinh: {x + a)ix + b)ix + c){x + d) + e = Vdi: a + b = c + d
Cach giai: Dat: t = {x + a)(x + b) = x^+(a +b)x + ab Khi do: (x + c)(x + d)^x^+(c + d)x + cd
^x^ -\-{a + b)x + ab-ab + cd = t-ab + cd
nen phu'cfng trinh trd thanh:
t(t-ab + cd) + e = 0^t^ + (cd-ab)t + e =
III PhiTrfng trinh: {x +af +{x +b)' + c =^Q
Cach giai: Dat: t = x + ^ - i ^ rdi du^a phu'cfng trinh da cho ve pJlu-dng trinh triing phu'cfng
IV PhtfcTng trinh: ax' + bx^ + cx' +dx + e=-0 (a ^ 0,e ^ 0) (vdi
'd\' e
a )
Cach giai: Chia hai va' cho x^, phu'cfng trinh trd thanh:
+ c =
2 ^ { d\
^b ;c + — bx]
•> CI
ex + Id
va phufcfng trinh trd thanh: at^ + bt ^ c - ~ = B&y la phiTcJng
b
trinh bac hai theo t 110
5.32: Giai cac phu'cfng trinh: 1) x' - x ^ + =
2) x ' - x ' - =
3) (X - l)(x - 5)(x - 3)(x - 7) - 20 = 4) {x + \){x + l){x + 3)(x + 4) - 35 = 5) x'+{x-A)'=%2
6) / + x ' - x ^ + ^ + = 7) x ' - x ' - x ' + x + l =
Gl§i
1) x ' - x ' + =
ãDat: ô = x' (di^ukien r > ) Khidd: -At + = Q
t = \
«|, =
• Vdi: ? = l < ^ x ' = l<^x = ±l
(1)
• Vdi: f = 3<s=>x'=3<^x = ± V
Vay phu'cfng t inh cd bon nghiem: x = ±l;x = ± A ^ 2) x' - x ' - -
Dat: i = x' (dieu kie i t > 0) Khidd: (2) ^ - r - =
t = -\{loai) t = A
• Vdi: i = <i=^ x' = ^ X = ±2 Vay phu'cfng trinh cd hai nghiem: x = ±2 3) ( x - l X x - ) ( x - ) ( x - ) - =
, Ta cc': (3) ^{x- l)(x - 7)(x - 3)(x - 5) - 20 =
(2)
(57)Dat: f = (;c-l)(;c-7) = ;c'-8x + Khido: (x-3)(x-5) = x^ ~Sx + \5 = t + Dod6: (3)^r(r + 8)-20 =
^ +St-20 = r -
; = - l
• Y6i t ^ ^ x^ - Sx + l =
^x^-Sx + = 0^x = 4±^ • vdi f = - ^ x ^ - j ; + = -10
<^x^-8;c + = -10
<^;c^-8;c + 17 = (pt v6 nghiem) Vay phifdng tfmh c6 hai nghiem: x = 4± VlT
4) (;C + 1)(X + 2X;C + 3)(A; + 4)-35 = (4)
Tacd: (4) ^ (;c + l)(;c + 4)(x+ 2)(x + 3) = Dat: t = (x + l)ix + 4) = x^+5x +
Khidd: (x + 2)ix + 3) = x^ +5jc + = r + Dodo: (4)<i=^;(i + 2)-35 =
<^f' + ; - = t = -l
t =
• V6i t = -l ^ x^ +5x + l\ (ptv6 nghiem) • Y6i t = <^ x^+5x-l = <^ X — -5±^/29
Vay phiTdng trinh cd hai nghiem: x = 5) x' +ix-4y =82
0 -
Dat: t = x + = x-2
-5±yf29 (5) Khi dd x = t + 2
x-^4 = t-2 112
nen (5) <^ + 2)'+ (/- 2)' - 82
^ + / + 6r\ + 4.r.8 + 16 + - / + 6r'.4-4./.8 + 16 = 82 ^2t' +48?' +32 = 82
^ t ' +24f'-25 = (*) Dat X = t^ (X> 0)
thi ( * ) ^ ^ ' + Z - = \X = l
^\x^-25(loqi)
Vdi x = i<^f' =i<^f = ±i • Vdi f = l<;=>;c = l + = • Vdi f = - 1 A: = -1 + =
Vay phiTdng trinh cd hai nghiem: x = ; x = I
.) JC'+5;C'-12A:'+5A; + = (6)
De thay x = khdng la nghiem ciia phiTcfng trinh (6)
Vdi x^O, chia hai ve cua phifcfng trinh (6) cho x^, ta diTdc: I
(6)^x^ +5;c-12 + - + — =
X X
n + x +
-X
-12 =
Dat: t = x + -.Th\: =x' + \ 2<^x' + \ e-2
X X X
Dodd: (6)<i^a'-2) + 5r-12 = <^<'+5r-14 =
t = -l f =
• Y^i t = -l ^ x + -=-1
X
<^ A:' + x - l =
<!4> J: = • -7±3N/5
(58)• Vdi t = 2^ x + - =
X
<^x^-2X + = 0<!F^ x = \
Vay phifcfng trinh cd ba nghiem: x-\ x = - ± ^
7) x'-3x'-6x'+3x + \^0
• De thay x = khdng la nghiem ciia phiTdng trinh • V d i x^^O, chia hai ve ciia phuTdng trinh cho x^ ta diTdc:
; c ' - ; c - + - + -!- =
X X
X ' + \ -
X \2 X
X
-
1 - X
x^
r X — - - =
- =
1
Dat: t = x phiTdng trinh trd - 3; - =
X
=-\\t^ = v i a - b + c =
• t = - tacd ; c - - = - l <i4> A : ^ - l = -;c,
X
^ x'^ +x-\
- i + Vs - - V = ,
1
• t = t a c d j : = - ^ x ^ - l = x < ( ^ x ^ - ; c - l = :
X
^x,=2^S\x,=2-S
Phifdng trinh cd bon nghiem la
z l ± ^ z i ^ ; + 75;2-V5
De 5.33: Giai cac phifdng trinh sau: ' x _ l
.3 X
1) l ^ ^ + ^ - i o 114
2) (4A: + l)(3x + 2){\2x -\){2x + 2) - =
3) 4(JC + 5)(;C + 6)(A: + 10)(X + ) - X ' =
Giai
1) Dieu kien x ^ T a c d : x ^ + ^ =
3 x^
x 3 X
Phirdng trinh trd thanh:
X '
^ ix' 16' = 10 ' X A — ^ + - T = 10
13 x^ 9 x\ >3 X,
x' 16 • x' 16 2
+ / + -
9 x'
3 = / ^ ^ ' - / + = 4^t = 2,t =
-3 * V d i ; = < ^ - - - = < ^ x ' - = 6x
3 X
x ' - x - = < ^ X = ± >/2T * V d i / =
-3
X 4
4^ =
-3 x -3
^ x^ - = 4x <^ x^ - 4x - =
<i=^A: = 6,x = -
Vay nghiem ciia phuTdng trinh l a : x = 6,x = - , x = ± ^ITA 2) Dat / = (4x + l)(3x + 2) = 12x' +11X +
thi (12x - l)(2x + 2) = x ' + 22x - = 2f -
nen phufdng trinh (4x + l)(3x + 2)(12x - l)(2x + 2) - =
2/' - / - = r = - l , f = * V d i t = -1 <^ 12x' + I x + = (v6 nghiem)
V d i t = <^ x ' + l l x - = Q < ^ x = ' ^ 24
-\\±yl2ri
Vay nghiem cua phiTdng trinh la x =
(59)^ 4(x + 5)(x + I2)(x + 6)(x + W)-3x^ =0
^4(x^ +nx + 60)(x'' +l6x + 60)-3x^ = Dat t = x^ + l6x + 60^ x^ + nx + 60 = t + x nen phiTcfng tfinh trd thanh: 4(t + x)-3x^ =0
^4t^+4xt-3x^ =0^(41^+4xt + x'^)-4x^ = ^ {It4-xf -{2xf =0^{2t + 3x)(2t-x) =
3 X
<^ t = x,t = — 2
*\6\t = - ^ x'+16;c + 60 = - ^ ; c ' + ^ + 160 = v6 2 nghiem vi A <
> A:'+16X + 60 = -—2 ^2JCN
-35±N/265 *Vdi / = - - ; c ^ A:'+16X + 60 = - — ^ J C ' + ^ + 120 = 2 • Vay nghiem ciia phu'dng tnnh la x = — ^ V265^
5.34: Giai phifdng trinh: {x^ +lx + \ +1 IA: + 28)-72 = Giii
Tacd: (x^ + 7A:+ 10)(A:^ +1 IJC+ 28) - 72 = (*) ^{x" +2X + 5A: + 10)(X' +4JC + 7X + 28)-72 =
x{x + 2) + 5{x + 2)][;c(jc + 4) + l{x + 4)] - 72 = ^{x^ 5){x + 2){x + l){x + 4) - 72 =
^ (x + 2)(A: 4- 7)(x + 4)(x + 5) - 72 = Dat: < = + 2){x + 7) = + 9x +14
Khidd: (;c + 4)(jc + ) - + x + 20 = / + Dod6: (*)<^f(/ + 6)-72 = +6/-72 =
/ = -12 116
• Vdi f = -12-4^;c^+9;c + 14 = -12
4 - + 9x + 26 =
Phifdng trinh v6 nghiem vi: A = 81 -104 = -23 < 0, ã Vdi ô = 44>x^+9A: + 14 =
A:^ +9x + 8'=0
A: = -
Vay phu'dng tnnh cd hai nghiem: x = -1 ; x = -
m 5.35: Giai phu'dng trinh: x' - 4;c' -19x' +1 06A: -120 =
Gi§i
J Tacd: jc'-4A:'-19A:'+106x-120 =
^x' - Sx" + A r ' + A:' - 5A:' + 6A: - 20A:' + 100A: -120 =
A:'(JC' -5A: + 6) + A:(A:' -5X + 6)-20(A:' -5A; + 6) =
(A:'+ A: - 20)(x'- 5A: + 6) =
^{x'' -4A: + 5A:-20)(X' -2X-3A: + 6) = ^ \x{x - 4) + 5(X - 4)][A:(A: - 2) - 3(A: - 2)] =
(AT + 5)(x - 4)(A: - 3)(A: - 2) = x + = [A: = -
jc-4 = 0 A: = x - = 0 A: = x - = 0 A: =
Vay phu'dng trinh cd bdn nghiem: x = -5;x = ; x = 3;x = 5.36: Phan tich nhan tiJ : A = x' - 5A:' +1 OA: +
x" + Ap du ng: Giai phu'dng trinh — = 5A:,
Giii Tacd: A = x' -5A:' +10A: +
(60)= x' -x'-2x^-4x' + 4A:' + 8;c -2;c" +2x +
^x^(x^-x-'2)-4x{x^ ~x-2)-2ix^-x-2) • ={x^ -x-2)(x^ -4x-2)
^(x^-2x + x-2)ix^ - J C - )
= x(x-2) + (x-2)](x^-4x-2) = (x-2)(x + \)ix' -4x-2)
A p dung: D i e u k i e n x^ —2 ^ x ^ ±yj2 x' +
Ta c6:
x'-2
^5x ^ x'+4 = 5x{x'-2)
^x' - ; c ' + x + =
^ (jc - 2){x + l){x^ - 4x - 2) -
x-2 =
<^ ;c + l = 4=^
x^-4x'2 =
Cac gia t r i ciia x t i m dUdc deu khac ± V a y nghiem cua phUdng trinh da cho la:
x =
x = -l x = ± S
= 2\x^ = -\;x^ - + ^/6;x, - - V
5.37: Cho biet phUdng trinh: {x + \){x + 2){x + 3)ix + 4) + m = CO nghiem la x^,x^_,x^,x^. Hay tinh M — — + — + — + — theo m
Gi§i
T a c : M = : ^ + ^ l ± A
x^x,
PhUdng trinh dUdc viet thanh:
{x + l)(;c + 4).(;c + 2){x + 3) + m = Dat t^{x + \){x + 4) = + 5x + thi (;c + 2)(;c + 3) = x ' + ; c + = / +
PhUdng trinh trd thanh: / ( / + 2) + m = ^ + 2/ + m = (*) 118
G o i t^,t^ la nghiem cua (*) thi ta c6: t^+t^= - , / / ^ = m
* V d i / = t, thi jc' + 5;c + = <^ + 5JC + - =
Theo dinh l i V i e t , ta cd x^+ x^= —^,x^x^ =4 — t^
* V d i r = thi x ' + 5JC + = ^2 A;^ + 5x + - ^2 = Theo dinh \i Viet, ta c6 x^+ x^ =^-5,x^x^ =:4 — t^
Do do: M = — — = ^ = - 4 - t , - t ,
%-{t,+t,)
1
+
4 - t , - t •2 J
= -5
= - - ( - ) - - ( - ) + m m + 24
f)i 5.38: Cho phu-dng trinh: x' - 2(m + 2)x^ = 0. T i m m de phUdng trinh cd bon nghiem phan biet
Gi^i
x' -2im + 2)x^ =0 (1)
D a t : / = A ; ' ( ? > ) thi (1) < ^ - ( m + 2)r + m ' = (2) IjPhu'dng trinh (1) c6 bon nghiem phan b i ^ t
<^ phUdng trinh (2) c6 hai nghiem phan biet diu dUdng A ' > [(m + ) ' - m ' >
S>0 2(m + ) > 4/n + > [ m > - l
m + 2>0 [m>-2
(61)§ PHLfONG TRINH CHLfA GIA TR! TUYET D6\
PHJCNG TRiNH CHLfA CAN THLfC
I Phifrfng trinh chiJa gia trj tuy^t do'i: 1) Dinh nghia va tinh chS't:
X ne'ux>0 —X ne'uX <0
. V7=
2) PhiTcfng trinh chiTa gia tri tuyet do'i:
f(x) = g{x)
m = g(x) fix) =gix)^
[fix) = -gix) 'gix)>Q
f(x) = gix) fix) =^-gix)
Nhan xet: Cac phiTdng trinh tren cd the bien doi nhuf sau:
• \f(x)\ \gix)\^fix) = g\x)
\gix)>0
fix) = gix) <^
fix) = g\x)
II Phufofng trinh chiJa can thiJc: 1) PhiTdng trinh chiJa mot can:
r- \B>0
A = B^ A = B
A > V i >
Ghi Chu: Neu phifcfng trinh cd dang: afix) + b^fix) + c =
Dat: t = 4fix) ; (/>0)
2) Phifdng trinh chiJa nhieu can: Phifdng phap:
4A = 4B ^
120
• Dat dieu kien
• Bia'n doi de hai w€ deu khong am • Binh phiTdng hai ve de khiJ bdt can • Riit gon de diTa ve dang cd ban
• Giai va ket hdp vdi dieu kien de chon nghiem Ghi chii: * Neu phifdng trinh cd dang:
g(x)
fix) + c =
Dat: / = / ^ ; ( / > ) \gix) * Neu phu'dng trinh cd dang:
V A + V A ^ + V A - V A ^ = B
Ta dung phep bien doi:
* A + V A - 1, = (/\ 1) + V A - + = ( V A ^ + I ) ' * A - V A - I = ( A - ) - V A - +
. =(vr 1 -
5.39: Giai cac phu'dng trinh: 1) jc-5 = •
2) 2 X - 3 - x + \ =
3) x-l =2x + l
:) Tacd: x-5 = 3^
Giai
;c-5 =
x-5^-3 x =
(62)2x-3 = x + \
2x-3 = - x - \ 4^ x = 3x = x^4
2
X = —
Vay phifdng trinh c6 hai nghiem: x = va — j • 3) Tacd: x-l =2x + ^ 2x + l>0 x-l = 2x +
x - \ -2x-l 4=>
3x = -6
Vay phiTcfng trinh c6 nghiem: x- -
9 ~
x = -
3x = -2 ^ x = -2
De 5.40: Giai cac phiTdng trinh:
1) x - x - = 2) x ^ - A : - A : - =
3) I J C- - - 2 = 4) x^ -5 x +6 = 1) * Xet ;c < phu-dng trinh trd thanh:
~x + x-2 = 2^Ox = 4^xe0
* Xet < jc < phiTdng trinh trd thanh:
x + x - = 2<;=>2;c = 4<^;c = (loai) * X6tx > phiTdng trinh trd thanh:
x-x + = 2^0x = 0^ xeR
Do x>2
Ke't luan: Nghiem ctia phu^dng trinh la : ;c > 2) * Vdi ;c < thi = -X, x-2=2~x
Phu-dng trinh trd thanh: 2x' + 7x - 5(2 - jc) =
<^ x ' + 12x -10 = <;=> ;c = - ± Vi4 Vi JC < nen chi nhan A; = - - Vi4
122
* Vdi 0 < A: < thi |x| = 2| =
-Phu-dng trinh trot thanh: 2x^ -lx-5(2-x) =
<^ 2x^ -12x - = 0 - X - 5 =
1
^X = -(l±yf2\^ 1 \
Vi - ( l - V2T) < < < - ( + ylri^ nen khong nhan
\±yl2\]
1
x = —
* Vdi JC > 2: thi x-2=x-2
Phu-dng trinh trd thanh: 2JC^ - 7X - 5(X - 2) =
<i:^2x'-12jc + 10 = ^ J: = 1, jc =
Vi JC > nen ta chi nhan x = ;
Ke't luan: Nghiem la x = 5, jc = - - Vl4
3) Taco: l x- - 1 - 2 = ^ =
x-2 x-2
- = hay x-2 = 3(v« x-2
x-2 - I
- l = -
— ——2<;4-Jc—2=—1 vd nghiem
< ^ X - 2 = ±3 4=^ X = 5,jc = -
Vay nghiemla: x = 5, x = -1
4) Dat/ = |x| (/>0)thi: jc^ - 5|JC| + = <^ - 5/+ =
^ t = 2,t =
* Vdi t = thi |jc| = 2 ^ JC = ±2
* Vdit = 3thi |x| = 3<^jc = ±3
Vay phu-dng trinh c6 nghiem la: x = ±2,x = ±3
De 5.41: Tim m de phiWng trinh x + •2x + m = CO nghiem
Gi^i
(63)-x>0
(x^ -3x + m)(x'^ -x + m) = x<0
x^ -3x + m = x<0
(x^ - 2x + mf -x"^ =0 x<0
(1) hay x^ -x + m = x<0 (2) * X a t m > 0: (1) c6 S = > 0, P = m >
(2) C O S = > 0, P = m >
nen chiing c6 nghiem thi cac nghiem deu du'dng va do ta't ca deu bi loai vi dieu kien x <0
* X e t m = : thi (I) ^ x^ -3x =^0 ^ x = 0,x =
(2) ^ x''-x^O^ x^O,x^l V i dieu kien x<0 nen nhan nghiem x =
* X6t m < 0: (1) C O P = x^x, = m <-0 nen < <
Va do phu'dng trinh c6 nghiem la x^
Ket luan: Phufdng trinh c6 nghiem m < 5.42: Giai cac phu'dng trinh:
1) i x^-Ax + A= A9 2) 4x^A^x-\ 3) Vl - 2JC' = ;c -
Gi^i
1) T a c o : ^x Ax + A =A9 ^ 4^x-2f = ^ x-2 =
A ; - =
2) T a c : ylx + \=x-\^
X- = - x-\>0
x^-\ {x~\f
x = 5\ x = -A7
124
<^ x =
•3) V l - x ' = j c - <^
4^
x > l
x + = - x + l x>\
x = V x = x - l >
I - x ' = ( x - l ) x > l
1-2A:' - 2X + x > l
X = O V J C =
-3
x>l
x^ -3x =
J C>
3X' - 2A: =
Vay phufdng trinh v6 nghiem
5.43: Cho phiTdng trinh c6 an x : yjx'^ - = x - a 1) Giai phu'dng trinh vdi a =
2) Giai va bien luan phu'dng trinh theo tham s6 a
GiSi
1) K h i a = : phu'dng trinh trd 4x^-A =x-2 \x-2>0
V x ' - = x - 4=^
x''-A = {x-2f x>2
x^-A = X^ - 4A : + x>2 \x>2 4x = 8 \ix^2 Phu'dng trinh c6 nghiem la x =
» : - a >
-A = (x-af
<^ jc =
2) ^lx^-A=x-a
(64)x> a
• a = ta CO (*) ^
• a ^ ta CO (*)
x> a
lax = a^ +4 x>0
Ox =
X > a
xe0
X = — a' +4
la
G i a i d i e u k i e n ^^—^> a ^ a>
la la
4-a^
la ~
a > ; 4-cr>0
a<0 • 4-a" <0 a>0 ; a^ <4 0<a<l
a<-l a<Q ; a ' >
T o m l a i :
• N e u < a < hoac a < - l : PhiWng t r i n h c6 nghiem a ' +
X =
la
N e u - < a < hoac a > 2: Phtfcfng t r i n h v n g h i e m
B e 5.44: G i a i phu-dng t r i n h : + x + = l^Jlx^~T^
D i e u k i e n :lx^-3>0^1x>-3^x> ~
T a c d A:'+4Jc + = 2^y2IT3
.2
^ AT'+ 4A: + -2 V 2^ + =
^ + 2A: +1) + |(2A: + 3) - 2V2^T3 +1 ^ U +1)' + (V2;c + 3 -1)' =
=
126
4=>
44>
(;C + ) ' = '
V2A: + - =
x + =
V2A: + - =
A: = -
V2A: + = 4=^
A: = - 2A: + =
< ^ Ar = -
N g h i e m cua phufdng trinh la x = - p 5.45: G i a i cac phu'cfng t r i n h :
1) A: - 5V X- =
2) 3A:^ + 2A: = lyjx^ +x +\-x
3) A : '- x - = yjlx^ - x + 12
1) A: - 5V^ - = ( )
D a t : t = ylx ; (t>0) • K h i d d : ( l ) ^ r ' - / - = 4=!>
[/ =
Ydi l = 6^4x = 6^ x = 36
2) 3A:' + A : - V ? T^ + -A: ^ 3U ' + X ) - ^ ^ ^ T^ - = 0(2)
D a t : t = y[x^~+x ; ( / > ) K h i d o : ( ) 4^ 3/' - 2? - =
t = l
1
' - « V d i / = l< ^ V ^ + A: = l<;=>A:'+x = l
- - V •i + Vs
<^A:'+A: - l = - < i = >
A: =
x —
V a y p h i cfng t r i n h c6 h a i n g h i e m l a : x =
1 - 1±
2) T a c : x'-4x-6 = ^lx'' - x + 12
(65)-4;c + 6)-12-V2(x^ -4A: + 6) = (3) Dat: t = ^2{x' -Ax+ 6) ; (i>0)
Khido: ( ) ^ y- 2- r = ^ f ' - f - = r =
' = - (/)
* Vdi f = 6^ ( X^ - 4A: + 6) =6<^2(X'-4A; + 6) = 36 <^;c^-4x-12 = 0<(=> '^"'"^
l;c=:6
Vay phiTdng trinh cd hai nghiem la: x = - 2 va x = De 5.46: Giai cac phifdng trinh:
1) ylU-x-Jj^=^2 2) V5A; -1 - V3x - 2 - V;c - = 1) Tacd: V l l - x - V ^ =
Dieu kien: l l - x > x-l>0 x<\\ x>l < ^ l < ; c < l l
( ) (*) Khi dd: (1) Vl - ^ = 2 + VA: -1
<^ 11 - ^ = + + (;c -1) <^ = - ;c 4 - X >
2 V I ^ = -X ^ 4 U - l ) = ( - x f x<4
x<4
4x-4 = l6-Sx + x^ x<4
x = 10\/x = ^x = (thoa dieu (*))
Vay phiTdng trinh da cho cd mot nghiem x = 2) ^5x-l-fix 2-^f^ =
x" -12A: + 20 =
(2) 128
Di^u kien: 5x-\>0 3x-2>0^ ; c - l >
x>-~ 2
x> — <^x>\ ~ x>\
Khi dd: (2) ^ yl5x-\ y/3x-2 + ^x-l
<!^5A:-1 = (3A:-2) + (;C-1) + 2V(3A:-2)(X-1)
x + 2>0 ^2yji3x~2Xx-\)=x + 4:^
44>
x>-2
12X' - 12;c - 8A: + = + 4x + 2 A: = 2V X = —
A; = 2V;C = — 11 11
4(3;C-2)(;C-1) = (X + 2)' x>-2
l l x ' - ^ + =
44>
So vdi dieu kien (* *) ta chi nhan nghiem x = Vay phiTdng trinh da cho cd mot nghiem x = Bi 5.47: Giai cac phu-dng trinh:
y/x-\ + ylx-5
Gi§i
(66)ix + \f>4 x > \
x + l>2 hoqc x + \ < - x>l
_ sJx^+2x-3
W
V d i d i l u kien do, ta c6:
y/x-l ylix+3)(x-l)
^ x > \
= + x
= + x
• = x + 3<=^^Jx + 3=x +
4=^ 'x + 3>0
x + = (x + 3Y 'x>-3
x^ +5x + = ^ x = -3Vx = -2
So vdi dieu kien (*) =^ phu'dng trinh da cho v6 ng liem
2) Dieu kien:
'x>-3
x + 3 = ;c^ + i +
;c>-3
x = -3\/ x ^ -
x - >
3 + ylx-5^0 <^ x>5
Ta cd:
^ 5 - ^ ^ = ^ ^ - ^ ' - ' ' ^ ^ ' - ^ ' ^ ^
^ _ ( - ) ( - V I ) ^ ^
9-A: + 4=^ VJ:-5+3-V;C-5 =
OyJx-5 = pt dung vdi moi A: > Vay phu-dng trinh cd nghiem y x>5
De 5.48: Giai cac phtfdng trinh:
1) ^Jx + + 4yfI^ +^Jx + S-6^JJ^-5 = 130
2) ^x + 2^x-[ +-ix-2^x-\ x-\-3
GiSi
1) Tacd \lx + + 4^x-l+^jx + S-6s/x-1-5 =
^ V(-^-1) + 4 V ^ T 4 + V(A: - ) - 6V ^ + - < =
f V ^ + 2f + j f V I^ - V - 5 =
<i=^ V x- H- + - - • Dieu kien Jc >
- 5 = (1)
• Vdi A;> 10(a) : thi - > N/9 = n e n V x - l - 3> va (1) <!=^VA:-1+2 + V A: - 1- - 5 =
<^ V x - = 3< ^ x - l = 9< ^ j ; = 10 (thda dk (a)) • V d i l < j c < : t h i V ^ < n e n V ^ - 3< v k
(1) V ^ ^ + 2- ( ^ / ^ - ) - 5 =
< ^ - = 0<i^0 = ( luon diing)
Dieu chiJng to moi x E I,10) deu la nghiem
Ket ludn: < x < 10 deu la nghiem ciia phtfdng trinh da cho *' 2) Tacd: + ^ / ^ + - V ^ ^ = ^ ^ i ^
^ 7(^-1)+-2V^^ + + VU-1)-2N/7^ + 1 = ^ i ^
- i + i ) ' + ^ | ( ^ ^ T ^ =
- + + - -
A: +
x + (2)
• Dieu kien: x >
• V d i A; > 2 (a) : thi V ; c-1 > =^ - 1 >
(2) ^ ^ / I ^ + l + V I ^ - l =
(67)2VA: - - x +
4=>
\e{x-\) = {x + 3f x>-3
-10A: + = :
x > -
1 X - = A:^ +6;c + ; c > -
x = ^x = (th6a dk (a))
V d i l < < {b) thi V ^ < nen V ^ - K O d o d d ( ) ^ + + l =
^ 2 = ^ ^ j i < ^ A ; = l ( t h a d k ( b ) )
T d m lai phtfdng trmh da cho cd hai nghiem: x = va x =
5.49: Cho phiTcfng trinh cd an x :
^x^-lx + \=yj(i + A^-4^-A4i
1) Rut gon v6' phai cu a phifdng trinh
2) Giai phu'dng trinh
GiSi
1) T a c d : + 4N^ = ' 4 + ^ + ( N ^ ) '= ( 2 + ^'
Tu'cfngtif 6-A-J2 =(l-yj2f Ta cd va'phai
= + > y | - | - N ^ = + V - + >y2 {vil> y[i)
2) yjx^-2x + l=yl6 + 4.j2 -yl6-4yf2
<^^[x-\f =2yf2
132
- = ^ ^ x - + ^ X = \-2yj2
Phu'dng trinh cd hai nghiem la: x = \ 2^2 \x = \- 2yf2
5.50: Giai cac phu'dng trinh sau:
\^ +4x +
1) c ' - ; c - + ,
+ 4x + 2 V A : ' - ^ - =
2) + = 5^{x-2){x-3)
1) Dat f = - A ; - V x ' +4;c +
G i i i
thi t > va phu-dng trinh trd thanh:
/ + - = 2<i=>/^-2/ + l = < ^ f = l
t x^~2x —
^ —z = ^ x ' + 6x + = (v6 nghiem)
4x^+Ax +
Vay phu'dng trinh v6 nghiem
) Nhan xdt x = 2, x = khong la nghiem
jgChia hai va' ciia phu'dng trinh cho ^[x - 2)(,x - 3) thi cd:
6.6 + ( - f
^ ( - ) ( - ) ^ ( x - ) ( - ) =
x-3 x-2 , x-2
1
'x-3
<^ 6/ + - = (vdi t =
t V x-3
x-2 ,t>0)
1 > ^ r - / + l = < ^ / = - , / = -
x-3 6 x-3
U-2 x-2
(68)It"-3 x =
Vay nghiem 1^: * x =
* V — X = —^
3 -
- -
2-3.2^ _ 190 1-2' " 63 2 - ' _ 2185 1-3' ~ 728
Bi 5.51: Giai phiTdng trinh: ^(x + 1)' +4^(x-\f = ^ x ' - l
- _
Nhan xdt ring x = ±\g la nghiem
Chia hai va'cua phiTcJng trinh cho ^x^ - ta diTcJc: {x-\f ^ JC + , ,
x^-\ x-\ x + \ -6 =
<^/ + l - = (vdi/=3pi-!-) t \ x - \
<^f'-6r + = 0<^r = 3±V5
Vdi r = 3^ X-\-\ X + \
<^ r = ^ r x - t = x + \
x-\ x-\
^(t'-\\x = e+\^x = ^^ (3±V5)'+1
<^ x =
134
5.52: Giai phifdng trinh:
^ji^-Ax + n + V3;c' - x ' +28 = -3;c' +-6x +
pjTiffVng phap: X^t phifcfng trinh: VT = VP (*)
VT > K Neu tim diTcic so' K cho: ~ Khi : (*) ^ vr = K VP = K
VP<K
(tu-c cac dau "=" xay ) GiSi
Ta cd:
Dau xay ra: ^
^Ix" -4;c + ll + V / - x ^ +28
= ^l[x^ -2x^\) + + p[x' -2A : ' +I) + 25 = ^2(.X-1)'+9 + ^3(A;'-1)'+25 >79 + V25
= + =
( x - l f = br^-l =
Matkhac: -3A;^+6JC + = -3(A:^-2x + l) +
= - ( ; c - l ) ' + < Da'u "=" xay ra: <^ X - = 0 <^ X =
E)od6: V2;c' - x +11 + ^3.^' -6JC' + 28 = -3A:' + 6;C + V2A;'-4;C + 11+V3A:'-6A:'+28 =
(69)x = l
<=> x = \ PhiTdng tnnh c6 nghieir nha't x =
136
§ 4. PHLfdNG TRINH V6 | NGHIEM SO NGUYEN
pi 5.53: H m so' nguyen a cho phifcfng tnnh:
[x-a)[x-lO) + l = 0 CO nghiemdeu la so'nguyan
T a c : {x - a){x-\0) + l = ^ {x - a){x-10)=-I (*) Neu xEZ,aeZ thi ta c6 x - a e Z,x-10 E Z
x — a — \ — a = —l hay x-\0 = Do tiy (*) ta suy rang: hay
-1 [ ; c - = l a = x — \
x = hay
a x +1
X = n
nan : a = 8, a = 12
* V d i a = thi phuTdng trinh troi thanh:
(x-S)(x-\0) + l = 0^ x^ -\Sx + Sl = • x = * V d i a = 12 thi phiTdng trinh trd thanh:
{x-l2){x-\0) + \ 0^x^ - 2A: + 121 = <^ A: = 11 Vay a = 8, a = 12 la gia tri can tim
5.54: T i m nghidm nguyen cila phifdng trinh: 4xy + 2x + 2y = 2S
Giii
Ta cd: 4xy+ 2x+ 2y = 2^ ^ 4xy+ 2x+ 2y+ \ 29 ^ ( x + l)(2>' + l ) = 29
V i xeZ,yeZ nen 2;c + G Z, 2^ + € Z d6 phiTdng trinh trd thanh:
2x + l = l \2x + \ 29 hay
2 j + l = 29 [2>' + l = l
2x + l = - l f2v + l = - hay '
2)^ + = - ;?.;y + l = - l
(70)F)e 5.55: T i m nghiem nguyen cua phu'dng trinh:
2xy + x + y = S3
4^
hay
T a c d : 2xy + x + yS3 ^ 4xy+ 2x+ 2y = 166 ^ 4^;}^ + 2A: + 2>'+1 = 167 ^ (2x + ) (2^ + ) = 167
'2x + \ \A: + = 167
hay
2>' + l = 167 [2>' + l = l
2x + \ -l , f2x + l = - hay
2 j + l = - |23; + = - ^ ( x = 0,>'=:83) hay [x = S3,y = 0)
hay{x = -\,y = -U) hay (x = - , ) ' = - l )
D e 5.56: T i m nghiem nguyen dtfcfng cua phu'dng trinh:
[x^+4){y^+\){z'+25) = S0xyz (*)
Gi^i
Ta bie't rang v d i a > 0, b > thi a + > yfcib va d i n g thitc chi xay a = b
Ap dung ket qua tren, ta suy ra:
+4>2yj4x' =4x
/ + l>2V7 = 2x
( V I X > 0)
(vi y > 0)
z^+25> 2^/257 = lOz (vi z > 0) Nhan ve'theo ve ba bat d i n g thitc vuTa neu thi c6:
(x' +4)(y' +l)(z' +25)>S0xyz
Do dd:
; c ' + = 4x
y^ +\ 2y ^
z ' + = 10z
x' =4
/ = 1 4=>
z' =25
x = y = l z =
Vay X = 2, y = 1, z = la nghiem can tim
f)i 8.57: T i m nghiem nguyen cua phu'dng trinh: 4x' +Sx^y-4y + 3y^-n =
GJii Phu'dng trinh du'dc viet
thanh-4(x' +2x^y + y^)-[y^ +4> ' + ) - = ^4[x^+yf-{y + 2f =13
^[2x^+3y + 2)[2x^+y-2) = \3
hay
2x^+3y + = l hay 2x^ + y-2 = l3
2x^+3y + = -\ 2x'+y-2 = -l3
2 x ' + y + = 13
2x^ + y-2 = l .2
hay
2x'=23
y^-S hay
2x'=-\ y =
2 A : ' + 3y + = -
2x' + y-2 = -\ 2x'=-\5
y = 4
hay hay 2x^=9
y = -S
Vay phu'dng trinh khong cd nghiem nguyen
De 5.58; T i m tat ca cac so' nguyen x thoa: x^ + S = l^Sx +1
D i ^ u k i e n Sx+ 1> ^ Sx >-I ^ x > ~ ~ ~
Do xeZ n e n j c > O T a c d :
T a c d : x^ +S = l^lSx + \ +sf =hJSx+ \]^ ^ A: ' + 16A;'+64 = 49(8A: + 1)
<i=>x' + x ' - x + 15=:0
^ x^ -3x' +3x' -9x' + 9x' -21 x^ + 43JC' - 9A: ' + ;C' - 7X- 5J C + 15 =
<^ x'(x-3) + 3x^ ( x - ) + x ' ( j c - ) + x ^ { x - )
+ 9A : ( X- ) - (X- ) =
4^(x- 3){x' + 3x' + 9x' + 43;c' + 129 A: - 5) =
(71)x-3 =
x' + 3x' + 9x' + 43x^ + 29JC - = x =
A: ' + ;C' + 9A: ' + 3X' + 9J C- = (*) X e t phufdng trinh (*):
* x = khong la nghiem ciia (*) V I- * x ^ O t a c o ; c > O v a x e Z n e n x > l
D o d d : X' + 3A: ' + 9A; ' + 3A: ' + 9J C>
=^ (*) khong c6 nghiem nguyen khac Vay phufcfng tnnh chi c6 mot nghiem nguyen la x =
Bi : T i m ta't ca cac so' nguyen x, y, z thoa phu-dng trinh 3x^ + 6y^ + 2z' + 3yh^ - x - =
Gi^i
Ta c6: phu'dng tnnh da cho
^ 3(;c' + 2y' + yh'-6x-2) + 2z' =
=^ chia he't cho =^ z chia he't cho ^ z^ chia he't cho • X ^ t z' = : Phu'dng tnnh trd :
3A: ' + / - x - =
^ ( ; c - ) ' + / = 3
< ^ ( A: - ) ' + r = 1
D o d d : / < 1 = > / < ' = ' ; ^ ' * / = ' = » ( j c - ) ' = l l V ly !
* y' = l'=^[x-3f=3'=^x-3 = ±3
^ X = hoac x =
Phu'dng tnnh cd cic nghidm
(J: = 6;y = l;z = 0);(x = 6;y = - l;z = 0)
(x = O;}-= l;z = 0);(x = 0; J = - l ; z = 0) 140
* / = ' - ^ (A; - ) = V6 ly !
• X ^ t z' > : Ta cd
3A: ' + / + z ' + ) ; ' z ' - ; c - =
^ ( ^ - ) ' + / + z ' + 3>''z' = 33
* / > t h i z ' + / z ' > 2.9 +3.1.9 > 33 (loai) * / = thi ( x - f + z ' = 3
V d i : * z' = thi (x - 3)' = 15 (loai)
* =^z'>6^ =36
T a c d (A: - ' ) + 2Z' > 3 (loai)
T d m l a i nghiem nguyen cua phu'dng trinh la: (;c = 6;); = l;z = 0);(x = 6;^ = - l ; z = ) ; (;c = 0-y = l;z = 0);{x = 0;y = - l ; z = 0)
5 : T i m nghiem nguyen cua phu'dng trinh: x ' - x y + / =
Ta c d : x ' - 4; c > ' + / = 9 4> ( x- ) ; ) ' + J c ' =
< ^ ( | 2; c - > ' | ) ' + | A : f =
Trong dd: | - y] |x| Nhirng 169 = 13' + ' = ' + ' Do dd cd cac kha nang sau:
l A : - y l =
\x\0
I x - y l = I A: I = l A : - y l =
IA:I =
;c =
y = 13
A: = 13
y = 26
x = V V
y = - V
x = ^
y = -
A: = -
y = -
x =
y = 22 V
= - = - 2
x = - ^ v =
(72)\2x-y\5
I A I -
;c = 12 x = l2 x = -l2 V
y = l9 3; = 29 ^ - - I Q
De 5.61: T i m nghiem nguyen cua phu'cfng trinh:
2 " + " + ^ = 3 ( v d i x < y < z )
Gidi
T a c : " + ^ + ^ = 2336 ^ 2^ ( l + 2^"^+2^"^) = 2'.73 (1) V i + ^ - " + ^ - M a s6Me nen:
( ) ^ ^ = '
1 + 2^-^+2^-^=73
Ta c6: (2) <^ 2^"' ( l + 2'-' ) = 2\9 V i + ^ - M ^ n6n:
x =
( ) ^ 2'-' = 2' 1 + 2'-' =
y = x + z-y =
y — X —
2'-' =
> =
z^3 + y
y = \i
z = n
Vay phufdng trinh da cho c6 nghidm Ih.: (x = 5;>' = 8;z = l l )
(2)
(3)
De 5.62: T i m nghiem nguyen dtfdng cua phu'dng trinh:
2 ^ + l - / =
Giii
Ta c6: 2^ + - = ^ ' = -
^ ^ = ( y - l ) ( y + l ) (*) Tir (*) suy y - va y + la ufdc so ciia 2" (vdi xeN)
nen chiing phai CO dang: y - = 2',>' + l = ^ ( v d i < / < ' ; iJeZ) Ta cd: 2^ - 2' = y + - (); - ) =
= ^ ' ( ^ - ' - l ) =
142
IT 2' =
2^-' - = hay
2" = - = =^ (/ = 1,;- = 2) (con 2'-' = 31a v6 l y )
= va do 2" = = 2^ = = 2' Vay nghiem nguyen du'dng can tim la : x = y =
^ x =
0 5.63: T i m nghiem nguyen du'dng ciia phu'dng trinh:
(*)
1 1 ,
GiSi
Tuf (*) =^ ^ > l,y > l,z >l,t>l ( v i - ^ < , - ^ < 1,4- < 1>4- < 1)
x^ / z^ e
* Neu X = y = z = t = thi (*) thoa man
* Neu mot bon so' x, y , z, t cd it nhat mot so' Idn hdn thi:
1 1 1 1 ,
+ -^ + ^ + ^ <- + - + - + - < !
- ' / ' ' ~ ' 4
(Gia svl A: > = ^ X > ^ ^ < - y, z, t deu Idn hdn hay
X
b k n g n e n < i 4x^ z r - < 7, - < j )
Vay x = y = z = t = 1a nghiem can tim
(73)CHl/dNGVI: H E PHLfOfNG T R I N H
§ HE HAI PHUdNG TRJNH BAC NHAT HAI AN HE BA PHadNG TRINH BAC NHAT BA AN
I He hai phifrfng trinh bac nhg^t hai in:
ax + by = c a'x + b' y = c'
II He ba phvfrfng trinh bac nhfi't ba an:
La he phifdng trinh c6 dang:
La he phiTdng trinh c6 dang:
ax + by + cz = d a'x + b'y + c'z = d' a"x + b"y + c"z = d"
Bi 6.1: Giai cac he phircJng trinh sau bang
[3x- y =
phifcfng phdp cong dai s6':
'x + 3y = \0 x-2y = -5
1) Ta c5: • 2x + y = (1) i5x = \0 ((l) + (2)) [3x-;; = (2)^[3x-y =
x =
h =
Vay he da cho cd nghiem: (2, 1)
2) Tacd: < x + 3>^ = 10 (3) rx +<=> 3>^ = 10
-2y = -5 (4) [5j; = 15 ((3)-(4))
144
x = \0-3y x = l
y =
Vay he da cho c6 nghiem: (1,3)
Bi 6.2: Giai cac he phifdng trinh sau:
'2(x + >') + 3(x->') =
1)
(x + ;;) + 2(x->') = 2)
2(x-2) + 3(l + >^) = -2
[ ( x - ) - 2( l + ;;) = -3
1) Tacd: •
Glii
2(x + >') + 3(x-;;) = (Sx-y =
(x + y) + 2{x-y)^5
(1) t3x->; = (2)
2x + = - l ((l)-(2))
y = 3x-5
2) Tacd: •
Vay he cd mot nghiem:
2(x-2) + 3(l+>') = -2
X = —
2
13
2' 3(x-2)-2(\ y) = -3
O
(3) (4) 2x + 3;; = - l
3x-2^ = '4x + 6y = -2
9x-6y^\5
'I3x + = 13 ((3)4(4))
l6>/ = 9x-15 x = l
(74)<=>
6.3: Ta bie't rang: Mot da thiJc bkng da thitc va cW tat
c& cac hd so'ciia n6 bang Hay tim cac gia tri ciia m va n de da
thiJc sau day (vdi bien so' x) bang da thtJc 0:
P(x) = {2m + n- 5)y + {5m-n-9)
Ta c6: P{x) = ( V x ) { m + n-5)x + (S/w-«-9) = (Vx) '2w + «-5 =
[5m-n-9 = Q
'2m + n = (1) \5m-n = (2)
'7/77 + = 14 ((l) + (2)) 77 = 5/77-9
f/77 =
'/77 = /2 =
Vay:
6.4: Giai cac he phiTdng trinh sau bang phifcfng phap the:
^x-2y = -3 \3x + y = -5
1) 2x + 3>' = 2x + 3y = -S
1) (ly
Gi§i
x-2y = -3 (1)
2x + 3>^ = (2)
Tuf phtfcJng trmh (1), bieu dien x theo y ta dtfdc: x = 2y - Tha' vac phifcJng trinh (2) ta du'dc:
2(2>'-3) + 3>' = o > / = 14
<^ y =
[y = \y =
Vay he (I) cd mot nghiem: (1,2) 146
2) (//> 3x + y = -5 (3) 2x + 3>' = -8 (4)
TuT phu'dng trinh (3), bieu dien y theo x ta dufdc: y = The vao phiWng trinh (4) ta duTdc:
2x + 3(-3x-5) = -8<::>-7x = ^ x = -l
-3x - Dodo: (//) y = -3x -
x = - l y = -2 x = - l Vay he (II) cd mot nghiem: (-1,-2)
x-y 2x + y _^
6.5: Giai he phu'dng trinh: 17 6.5: Giai he phu'dng trinh:
^^ + ^ + ^ - ^ = [ 19
x — y 2x-\-y
7 17
Ax-\-y y —
= = 15 5 19 31x-10> = 833 76x +24^ = 1460 93x-30^ = 2499 95x + 30y = 1825
x = 23
30)' = 1825-95x x = 23
y = - \
4=^
17x-17>' + 14x + 7>' = 833 76x + 19>' +5)^-35 = 1425
31x-10)' = 833 19x + 6>' = 365 188x = 4324 95x + 30>' = 1825
x = 23 30y =.-360 Nghiem cua he phufdng trinh la x = 23
y = -n
(75)6.6: Cho he phufcfng trinh (m la tham so'): mx — y = \ —x + y = -m 1) ChiJng to m = he phufcfng tfinh c6 v6 so'nghiem 2) Giai he tren m ^
1) Khi m = he phufcfng trinh trci thanh:
Ox^O xeR y — x — l x — y = l
—x + y — —\ — x — \ He phtfcfng trinh c6 v6 so'nghiem
inx — y = l {mx — x = l — m —x + y = —m [—X -\-y — —m xim-\) = \-m \x = - \ y = -m-{-x [>' = - m - l He phifcfng trinh cd nghiem [x = — \\y = —m —.l) 2) Khi m ^ 1:
6.7: Cho he phufcfng trinh hai an x, y: —2mx + y = mx-\-3y = \ 1) Giai he phufdng trinh m =
2) Giai va bien luan he phufcfng trinh theo tham so'm GiSi
1) Khi m = he phiTcfng trinh trd : -lx + y =
x + ?>y = \
-6x + l,y = \5 x + 3y = \
-Ix^U jc + 3y = l x^-2 \3y = Nghiem cda ht phufcfng trinh Ik (jc = - ; y = l ) 2) T a c : ( " ^ - ^ + ^ = ^ + 3^ = 15
{mx-\-'iy = \ = \
x = -l - + j = l
x = -2 y = \
148
4=>
-7m;c-14
mx + 3y =
mx = —2 \ mx
* m = : Ta c6 (*) ^
Ox = -l
= +
xe0
' ^ H# phufcfng trinh v6 nghiem
x = - — m * m ^ : Ta CO ( * ) <^
1 - m - ^
y = m
m y = \
He phifdng tfinh c6 nghiem la
2_ m y = \
(*)
6.8: Cho he phufdng trinh: 2x + 3y = m
Sx-y = \ (1)
1) Giai he ( l ) k h i m = -3
2) Tim gia tri cua m de he (1) cd nghidm (x > 0, y < 0)
Giii
1) Khi m = -3 he phu-dng trinh (1) trd : 2x + 3y = -3
5x-y^\
2x-]-3y = -3
15x-3>' =
;c =
17A: =
15A:-3>' = 4^
-3y =
x = y = - \ Vay he phufdng trinh cd mot nghiem (x = 0, y = -1) 2) Giai he (l)tadu'dc:
2x^-3y = m 5x-y = \
2x + 3y = m Jl7.x = m +
15;c-3y = ^ 1-3>' = 3-15J:
(76)X = m +
17 ^
y = 5x-l
X = m +
17 _ 5(m + 3) ~ 17
X =
y =
m + 17
5m + 15-17 17
X —
y = m +
TT
5 m - 17 He (1) c6 nghiem thoa (x > 0; y < 0)
m + 5 m -
X = >
17 <
m + > 5 m - <
4^ m > - 5m < m > - 0 < m <
-m < J
Do dd - < m < - thi he (1) c6 nghiem thoa (x > 0; y < 0) 5 6.9: Cho he phiTcfng trinh mx + my =(l-m);c + ); = (2) —3 (1)
1) Giai he phiTcfng trinh m =
2) Tim m de he c6 nghiem (x < 0; y < 0)
Giii
1) Vdi m = 2, he phuTOng trinh trd : 2;c + 2y = -
-x + y =
x-\-y =
y = x
2x = ^ y = x
150
Vay vdi m = he phifdng trinh cd nghiem la: 3 3^
2) 4=^
mx + my = -3 {\-m)x + y =
mx + my = —3 y = -{\-m)x
mx -m{\-m)x = -3 y = {\-m)x
i^m-m + m^^x = - y {\-m)x
m 'x = -3
x<0 y = -{l-m)x
Nghiem thoa dieu kien
[y <^
Vay m > la gia tri can tim
< ^ l - m < = ^ m > l
Bi 6.10: Cho he phifdng trinh: mx — y = 3x + my = (1) 1) Giai he (l)khim=
2) Vdi gia tri khac nao cua m thi he (1) cd nghiem thoa man :
x + y = \ m
m ' + 1) Khi m = he phifdng trinh (1) trd :
x — y — 3x + y =
4x = l y = x —
_ _
^ ~ 1 J Vay m =1, he phiWng tnnh cd mot nghiem
2) Vdidi^ukien m^tO.tacd:
(77)mx — y =
3x -\- my = [m^ ^-?>)x = 2m +
2)X-\-my —
rri^x — my = 2m 3x + mj =
2 m + X = + 3.(2m + 5)
m ' + -\-my = X = 2m + +
6m + 15+ m(m^ +3)y = 5m^ +15 4^
X — 2 m +
m' +
m(m^ + 3\y = 5m^ — G;??
2 m + m ' +
(5m-6) 4=^ m(m^ + )
X = 2m+ m ' + 5m — m' +
He phiTdng trinh (1) c6 nghiem thoa: A: + y = m^
m ' + khi va chi khi: 2m + ^ m — m
m ^ + m ^ + m ^ + <^ 2m + + 5m - = m^ + - m^
4
44.7m = 4<^m = — 4
Vay m = — he phifcfng trinh (1) c6 nghiem thoa:
x+y=l- m' +
6.11: Gia su" h6 phifdng trinh sau day c6 nghiem: Chtfng minh r^ng = 3abc
ax + by = c bx + cy = a cx-\-ay — b
GiSi
Goi (j^o '>o) 1^ nghidm cua hS phu'dng trinh da cho 152
+ ^Jo = c (1) Tac6: hx^ + cy^ = a (2)
+a>o =Z7 (3) _
Nhan hai ve ciia (1), (2), (3) Ian tot vdi ,b^ ta c6 cb^x^+ab^y^=b'
=^a^+b^+c^ =[a^b + b^c + ac^)xQ+[a^c + ab^+bc^)y^ (*) Nhan hai ve' cua (1), (2), (3) Ian liTdt vdi ab, be, ac ta c6:
a^bx^ + ab^y^ = abc b^cxQ + bc^y^ = abc UC^XQ + a^cy^ = abc
=^3abc =(a^b + b''c + ac^)x^+[a^c + ab^+bc^)ya (**) So sanh (*) va (**) ta du^dc : a' +b' +c' = 3abc
6.12: Giai he pht/dng trinh: 'x^y + z=^\ x + 2>' + 4z = x + 3y + 9z 21
Giii
x + y + z = l (1) x+y+z=\
Taco: x + 2y + 4z = S ( ) ^ y + 3z = l ((2) -(!)) (4) x + 3y + 9z = 21 (3) y + 5z = 19 ((3) -(2)) (5)
x+y+z=l 'x = \-(y^z) x^6
mjt ^ y + 3z = l ^ y = l-3z y = -n
lz = \2 ((5)-(4)) z = e z =
(78)Nghi6m cila he phiTdng trinh la
x = z = De 6.13:
X y z
Giai he phtfcfng trinh : - -2 4x + 3y-2z = -\
Ta CO :
X ^ y _ z
6 - ~ - 4x + 3y~2z = -l
6 - y z - - 4x + 3y-2z
X = - z
y = 5z <^ z = - l
= -
'x = y = - z = -l
X = —3^
y = 5z
- z + z - z = -
Vay he phufdng trinh c6 nghiem l a : ( x = ; y = - ; z = l )
x + y = Sz-l
6.14: Giai he phiTdng trinh ^ y + z = ^4x-l
Z + x = ^4y-l
dm
Dieu kien: x,y,z>-
4
Nhan vao m6i phifdng trinh vdi r d i cong lai, ta diTdc: 4x-2y/4x~l+4y-2^4y-l+4z-2.j4z-\^0
154
{4x-\) -2V4JC-1 + l] + [ ( 4} ; - l ) - ^ ^ + l] + [ ( z- l ) - ^ ^ - + ( , y l ^ - i f + ( V > ^ - l ) ' + ( z ^ - i f =
=
V x - =
^4y-\=\^
V z - =
2
^ =
vay h6 CO nghiSm nhat 1 I
De 6.15: T i m m o i x, y, z phifdng trinh:
x + y + z + = 2ylx-2 + y J y - + V z-
GiSi
x - > [ ; c > D i e u kien: y - > < ^ y >
z-5>0 \z>5
T a c o : x + y + z +4 = 2ylx-2 + 4^y-3+6ylz-5
x + y + z + 4-2^f7^-44y^-6^[z^ =
f ( A : - - ^ / ^ ^ ^ + l ) + ( y - - y ^ + 4) + ( z - - ^ / F ^ + 9) = V ^ - l f+ ( A/ r ^ - ^ ) ' + ( V F= - ) ' = •
( V ^ - ) ( V ^ - ) ' = ( V ^ - ) ' =
^ ^ - =
^ V > ^ - = V F^ - = o
(79)- - X - = - = ^ ^ y - - = z - =
Nghiem ci5a he phu'cfng trinh la : [x = 3;y = 7;z = l4)
156 ,
§ H E GOM MOT PHadNG TRINH BAC NHAT
V A MOT PHLfdNG TRINH BAC CAO
I Dang: 'ax + by + c = 0 (1)
fix;y) = 0 (2)
trong d6: f(x,y) = la mot phu'cfng trinh bac cao theo hai an x va y •
I I Phifflfng phap:
1) TO phu'cfng tnnh (1) rut mot an x hoac y theo an lai 2) The vao phu'cfng trinh (2) de du-a phu'cfng trinh (2) ve
phu'cfng trinh mot an
De 6.16
1) ^
: Giai cac he phifcJng trinh sau: j c - v + =
+xy =
x-y =
x'^ +xy-y^ =1
1)
x-y + = 0 (1) x^+xy = 4 (2)
Giil
Tacd: (1)<^>' = A: +
The vao (2): x'' +x{x + 2) = 4^2x^ +2x-4 = <^ • V d i : X = thi y = + =
;c = l
x = -2 • Vdi: x = - thi y = - + =
Vay he c6 hai nghiem: (1; 3) ; ( - 2; 0)
x-y = 0 (3)
x'+xy-y^=\)
fc Ta c6: (3) ^ y = X th6' vao (4)
x^ +x.x-x'^ = x^ =1^ x = ±\ 2)
(80)• Vdi: X = - 1 I h i y = -
• V d i : A ^ thi y =
Vay he c ) hai nghidm: (-1; -1) ; (1, !).•
6.17: Cho he phifdng trinh: x + y = l
x' =m{x-y)
1) Khi m = 1: he trd thanh: 1) Giaihe m =
V d i gia tri nao cua m thi he c6 ba nghiem phan biet ? Giai
x ' - / = x - y (2) Ta C O : (1) ^ >'= 1 - A; Th6'vao (2):
x' -{\-xf =x-{\-x)<=^ x'-(\-3x + 3x^-x') = 2x 2A: ' - 3A:^ + X = <^ x(2x^ - 3A: + l ) =
x =
2A: ^ - 3X + = x = l ^y = 0)
2
Vay he da cho c6 ba nghiem: (0;1) ; (1 ;0) ; ( 2
2) T a c d : x + y = \
x' =m{x-y) x + y = l
{x-y)(x^+xy + y^) = m{x-y x + y = \
{x-y)[x^ +xy + y^-m)=^0 y = \-x
x-y^O v ( / / )
y = l — x x^ ^xy + y^
158
a Tacd: (1) <^
y = \ x
x-[\-x) =
Vay ha (I) C O mot nghiem:
y = 1 — X
U'2j
b Xethe (//)
y = \ X
1 <^
X = —
2 •
(3)
1
X — —
2
x^ +xy + y^ -m = 0 (4) Thay: y = - x tiY (3) vao ( ) :
x^ +x(l-x) + (l-xf -m = 0<F^ x'^ -x + l-m = (*) He da cho c6 ba nghiSm phan biet 4=^ (*) cd hai nghiem
phan biet khac ^
A > •l1 2
2 l2,
m > —
-4 , ^ m>-
3
4
44>
l - ( l - m ) >
m ^ —
Vay m > - h6 da cho c6 ba nghiem phan bidt
6.18: Giai he phiTdn^ tiinh:
x + y = A (1) / + / = (2)
( l ) ^ y = 4- A: T h a ' v a o ( ) : x' + ( - ; c ) ' = (*) Dat: t = X - <^ A: = / + Thi:
( * ) ^ ( / + y + ( / - ; =
(81)^ t' +4.t\2 + 6.t\4 + 4.t.S + l6 + t' ~ A.e + 6.r' - A.t.9, +16 - 82
<^2t' +48^^ - = ^t' +24r^-25 =
r = -25(1) Vdit= 1, tacd: x = ; y =
Vdit = - l , t a c d : x = ;y = Vay h6 c6 hai nghiem: (3, 1); (1, 3)
§ HE DOI XUfNG - HE DANG CAP
I d6'i xiJng lo^i I:
1) Binh nghia: H6 : • duTdc goi Ik M dd'i xtfng loai I n6u thay x bdi y ngifdc lai thi moi phiTdng trinh h6 khong thay ddi
2) Phifdng phap:
• Dat:S = x + y ;P = x.y
• Dura he da cho vi he c6 hai an S; P
• Tim S; P Khi dd: X, y Ik nghiSm ci5a phifdng trinh: X'-SX + P =
3) Chil v:
a) Dieu kien c6 nghiem: -4P>0
b) Neu (x, y) la mot nghiem ciia h6, thi (y, x) cung la mot nghi6m ciia h6
c) *
+ =(x + yy -2xy = S^ -2P
3P) - (x + y){x^ -xy + y^)
= (x + yiix + yy-3xy\=S.{S'
* =(x' + / ) ' - x V
= [(x + yy-2xy\
= [S'-2PJ-2P\ II H$ dfi'i xuTng lo^i II:
1) Bmh nghia: He:
• U(^,)') = G (2)
dtfdc gpi la h6 d6'i xiJng loai II na'u Khi thay x bdi y VJ» ngtfdc lai thi phufdng trinh (1) bien phufdng trinh (2) va phudng tiinh (2) bien phifdng trinh (1)
2) Phifdng phap:
• Lay (1) - (2) de difa v^ mot phiTdng trinh dang tlch s6' va
giai nghiem theo nghiem
_ • The vao mot phifdng trinh he de
(82)III Hf dang cfi'p:
1) Binh nghla: La he phiTdng trinh c6 dang: ax^ + bxy + cy^ = d
a'x^ +b'xy + c'y^ = d'
2) Phifdng ph^p: X6t hai trifdng hdp:
• THl: X = 0: Tha' trifc tiep vao he de giai
• TH2: x^O: Dat: y = kx
- Tha' vao he, khur x ta difdc mot phifdng trlnh theo k - Giai tim k, iJng vdi mSi trifdng hdp c\5a k ta tim dtfdc nghidm (x, y) ciia he
Ghi chu; a) C6 the xet hai trifdng hdp: y = va y^O b) Khi he phiTdng trinh c6 bac cao hdn ta van c6 th^
dung each giai tren n6u cac so'hang d ve trai ciia he R ding cap
6.19: GiSi he phtfdng trlnh sau: +y^ +x + y = S (1)
x'+y'+xy (2)
Giii
Day Ik mot he do'i xiJng loai I
Dat: S = X + y va P = x.y
Tsic6:* {l)^{x + yf-2xy + x + y = S
4=»5'-2P + - (1') * (2)^{x + yf-xy = l
<^S^~P = (2') TO (2') suy ra: P = 5' - The P vao (1') ta cd:
5 ' - ( ' - ) + =
<=^-S^ + S + \A = S^S^-S-6 = 0^S = 3,S = -2 • Vdi S = thi P = 2: X, y la nghiem ciia phufdng trinh:
X ' - X + = ^ X = 1,X = nencd {x = l,y = 2y,{x = 2,y = l)
• Vdi S = - thi P = - 3: x, y la nghiem cua phufdng trinh: 162
+ 2X - = <^ X = 1, Z = - nencd [x = l,y ^-3);{x =-3,y = \)
Vay h6 cd bdn nghiem gom:
{x = \,y = 2);{x = 2,y = l);{x^ l,y = -3);{x = -3,y = l),
f)i 6.20: Giai he phrfdng trinh: x^ +xy + y^ = (1) x + xy + y = (2)
(!') (2')
Gi^i
Day la h6 do'i xi^ng loai I: Dat: S = x + y; P = xy
Tacd:* (1) ^ {x + yf - xy4-* {2)<:^{x + y) + xy =
^S+P=2
TO (2') suy ra: P = - S The vao (1'): 5 ' - ( - ) = < ^ ' + - =
S = -3 [5 =
• Vdi S = - thi P = ; X, y la nghiem ciia phtfdng trinh: + 3X + = Phtfdng trinh v6 nghiem • Vdi S = thi P = 0: X, y Ik nghiem ciia phiTdng trinh:
X = X = n6n cd: (x = ; y = 2); (x = 2; y = 0)
Vay he da cho cd hai nghiem: (x = ; y = 2); (x = 2; y = 0) , X ' - X = 0<^
6.21: Giai he phufdng trinh: x'=3x + Zy (1) / = > ' + 2x (2)
GiSi
Day Ik he dd'i xiJng loai II
(\)-(2): x'-y'={3x + 2y)-{3y + 2x)
(83)<^{x- y){x + y) = x-y<?^{x- y){x + y-l) = x-y =
[x + y-\ • Y6i: x-y = 0^y^x.Th€wa.o(\)
x^ =3x + 2x x^ -5x = \x = (y = Q) ^[x = (y = 5)
• W6i: x + y-l = 0^y = l-x Th6'w^o{l): x"" =3x + 2{l-x)<:^ x^ - x - =
x^-\ = 2)
^\y = {y = -l) Vay he c6 bon nghiem:
(x = 0,y = 0);(jc - 5,y = 5);(x = -\,y = 2);{x = 2,y = -1)
Bi 6.22: Giai he phifdng trinh: x' =3x + Sy (1)
y'=3y + Sx (2)
Giii
Day la he do'i xiJng loai II
il)-(2y.x'-/={3x + Sy)-{3y + Sx)
( J : -y)(x' +xy + y') = -5{x-y) <^[x-y)[x^ +xy + y^ +5)^0
x-y =
x" ^xy + y^ +5 = Q
• Vdi: ;t - >- = <^ > = jc The vao (1): x^ = 3x + %x^ x^ - nx = Q
\x = {y = 0) O ;c = -VrT {y = - ^ )
x = VrT (}' = ViT) 164*
4 (v6 nghiem)
Tdm lai: He da cho c6 ba nghidm:
[x = ay = 0);(x = -^\,y = -yfu),(x = ^/^T; y = ^/^Tl
p i 6.23: Giai h6 phifdng tfrnh: ^ / = ; c ' - x ' + ; c (1)
^ ' = / - / + ) ; (2)
Giii
Day la he do'i xifng loai II:
i2)-iiy x'-y'=y'-x'-3(/-x') + 2(y-x)
^(x'-y^)-2(x'-y^) + 2{x-y) = ' ^{x-y)[x'+xy + y')-2{x-y){x + y) + 2{x-y) =
{x-y)[x^ + xy + y^ -2x-2y + 2) = x-y =
x^+xy + y^-2x-2y + =
• Vdi: x-y = 0<=^y = x.Thivko(l): x\=x'-3x''+2x^x'-4x^+2x =
^x[x^ -4;c + 2) = x = (y = 0) x = - ^ ( y - - N ^ ) x = + yf2 [y = + y/2] • Vdi: A:^+X>' + / - 2A: - > ' + =
2;c^ + 2xy + 2}'^ - 4x - 4}' + =
^ x^ +y^ +(^x'^ +y^ +4 + 2xy-4x-4y) = ^x' +y' +{x + y-2f =^0
(84)x =
^ y = (v6 nghiem) x+y-2=0
Tdm lai: He da cho cd ba nghiem:
{x = 0,y^0);(x = 2-^,y = 2-^);(x = + ^,y = + ^ ] f)i 6.24: G i i i cac hS phiTdng trinh:
1) /-3xy = 2) x^-3xy + y^ =-\
x^-4xy + y^=l 2) 3x^-xy + 3y^ =13
Giii
y^-3xy = x^ -4xy + y'^ =1
Day Ik h6 phifdng trinh ding cap (bac hai) 1) (/)
• Vdi y = 0: He (I) trd thanh: • Vdi y^O: Dat x = ky, tacd:
0 =
x'=l (khdng thda)
/{I-3k) = (1) / ( ^ ' - J t + l ) = l
y-3Jk/=4
ky-4k/+y'=\ TO dd ta cd:
1.(1 - 3/:) = ( i t ' - 4^ +1) 4=^ 4jt'-13ifc + = \k =
4 Vdi k = 3: Thd'vao (1), ta diTdc:
(v6 nghidm) ^-4
Vdi /: = 1 The vao (1), ta diTdc: y^ = 16
166
Vay he da cho cd hai nghiem: (x = l;y = 4);(;c = - l ; y = - )
x^-3;cy + / = - l 3x'-;c>' + y ' =
Day Ik he phiTdng trinh ding cap (bac hai) 2) (//)
Vdi X = 0: H6 (II) trd thknh:
3 / = (khdrg thda)
• Vdi x ^ 0: Dat y = kx, ta cd: x'-3kx'+k'x'=-l
3x'-kx^+3k^x'=\3
TOdd tacd: l3(\-3k + e) = -\(3-k + 3e) k =
<f=^2Jfc'-5^ + = 04^ k = —
2 • Vdi k = 2: The vao (2), ta duTdc:
x'(\-3k + e) = - l (2)
^2^3_k + 3e) = l3
x^=l<=¥ x = \y = 2.1 = 2)
x = - l {y = 2i-\) = -2) Vdi it = - : Tha' v^o (2), ta duTdc:
2 x^=4^
x =
x^-2 , = i.(-2) = - l Tdm lai he da cho cd bo'n nghiem:
(85)6.25: Gidi hd phiTdng trinh:
xy{x-y) = 2'
Giii
Day Ik he phiTdng tfinh dang cap (bac ba) • V d i X = 0: h6 trd thanh: =
[0 = • Vdi x^O: Dat: y = kx, ta c6:
(kh6ng th6a)
He 4^ x'-k'x' =1 kx^ {x-kx) =
x' (l-k') = x'k{\-k) =
T i r d d t a c d : ( = 7^(1-it) <^2^'-7Jt'+7^-2 =
^{k-l)(2e-5k + 2) = 0^ k = l k = k='-
2
• V d i k = l : T h ' v a o ( l ) , tadufdc:0.= (v6 nghiem) • V d i k = : T h ' v k o ( l ) , t a d U ' d c :
- 7A: ' = < ^A : = - (>' = ( - l ) = - ) • V d i A: = - : T h ' v a o ( l ) , tadifdc:
2
=S^x = y = -.2 = \
T d m l a i h ^ da cho cd hai nghiem: (x = -\,y = -2);(x = 2,y = l )
168
I
§ HE PHJONG TRiNH CO DANG DAC BIET
x — y — xy = 6.26: Giai he phifdng trinh:
x^ +y^ +xy = l x^ +y^ +xy = l
Gi^i
Dat:
Hd ^ u = X
v = -y
M + V + MV = + - MV =
(M + v) + MV = (M + v)^ — 3uv —
S + P = (1) ' - P = l (2) (vdi: S = u + V ; P = u.v)
T a c d : (1)<^P = 3-S
The vao (2): ' - 3(3 - 5) = <^ ' + 35 - = ^ ^\S^2 (P = l )
V V d i : S = ; P = thi u, V la nghiem ciia phifdng trinh
X^ - X + l= < i^ X = l < ^ M = l
v = l
x = l y = -\ • V d i : S = - ; P = thi u , V la nghiem ciia phiTdng trinh:
X ^ + A : + = phiTdng trinh v6 nghiem T d m l a i he da cho cd mot nghiem: (x = ; y = - 1) m 6.27: Giai cac he phiTdng trinh:
3
1) 2x — y x + y 1 !_
2x-y x + y I
= -l
(I) 2)
=
1
2x-y x-2y
_2 L_ = J
-2x-y x-2y
(11)
(86)1) D i l u k i e n :
Dat:
u =
x~2y^0 x + y^O
1 Gidi (*) V = 2x-y ^ thi: x + y
4^
3M- 6V = - M - V =
2x-y = x + y = Vay he da cho c6 mot nghiem: (x = 2; y = 1)
2x-y^0 x-2y^0
1 2) D i l u k i e n :
Dat:
V = •
2x-y I x-2y
thi: (//) <^
2M + 3V = -2 2M - V = —
18
4^ 2A: - > ' = 12 J: - > ' =
X =
y = V i y h6 da cho c6 m6t nghidm: (x = ; y = -2)
6.28: Gidi hd phtfdng tiinh:
;c + y + - =
{x + y)-= y
Dieu kien: y^Q u = x + y Dat:
V — thi he <!=>
M + V = M.V = Do dd: u , V la nghiem ciia phiTdng trinh: 170
- 5X + = <J=> X =
M = V = V
V d i :
V d i :
M = V =
x + y =
^ = x = 2y
« = v =
+ y =
^ =
A; + >' = X = 3^
M = v =
jc = y = l
_
1
T d m lai he da cho c6 hai nghiem: [x = 2;y = l ) ; ' ^
) l 6.29: Giai he phufdng trinh: x - y = [4y-4^){\ xy) x' +y^ = 54
GiSi
•^-y = (V>'-V^)(i.+ ^ ) (1)
x ' + / = (2) Dieu kien: A ; > ; y >
• Na'u: x>y ^ 4x> d o d d : x - > ' > O v k ( y - V ^ ) ( l + ; c y ) <
n6n (1) sai Vay x > y khong thda • N6ii: x<y=^yfx <^Jy d o d d :
x - y < va -4x){\ xy)>Q n6n (1) sai Vay X < y khdng thda • V d i : x = y
Tilf (2) ta cd
x' +x' =5A^2x' =5^^x' =21^x' =3' ^ x = {y = 'i)
I V d i X = ; y = thda phiWng tnnh (1)
(87)6.30: Giai he phiTcfng trinh: {x + yf -4{x + y) = l2
{x-yf~2{x~y) =
G i i i
{x + yf-4{x + y) = \2 (1)
{x-yf-2{x-y) = 3 (2)
Datu = x + y ; v = x - y
T i r ( l ) t a c : - 4M = 12 - 4M- =
u =
u = -2 • Vay
TO(2)tac(3: - v = <^ - v - =
x + y = x + y = -2
v = - l
v = • Vay:
(1)
(2)
TO(l)va(2) tacd:
x + y = 6,x-y = - l x + y = 6,x-y = x + y = -2,x-y = - l
:^+>' = - ^ - l =
He phiTcJng trinh cd bon nghiem la:
(5 7](9 3]( I] (I
'x-y = -l\ x-y = 3 j
- -1
''~2'^~2
9
x = —;y = —
_ _ , _J_ ' - ^ "
1
x = —;y = —
2
U' A ' J' i ' j ' i ' j
-6.31: Tim ta't cd cdc gia tri x, y thoa he: ^ ^ + / <
GJii
'x'+y'<l (1)
x'+y > (2)
172
( ) ^ x' ( A- ) <
y'{y-\)<0
x'{x-l) + y\y-l)<0 (*) x' < I ;c <
/ < 1 (3)
(1) vk (2)
= » A : ' ( A: - ) + / ( } ' - ) > ( * * ) :
TO (*) va (**) ta c6: x' {x-l) + y^ {y-l) = '
Ket hdp vdi (3), ta diTdc: ,
x' {x-l) =
/ ( y - l ) =
Thu" lai, nhan thay
x^OV x-\ y=Oyy-\=0
x = 0, *
hoac
x = l , x = 0, *
hoac thdahe: .>' = • y =
x = 0\/x = l
y = Oyy = l, • ?
X* +y^ <1
x'+y'>l
Bi 6.32: Giai he phiTdng trinh: {x-yf+3{x-y) = 4 (1)
2x + 3>' = 12 (2) G i i i
Dat: u = x - y Ta cd:
(1)<^M'+3M = < ^M' + 3M- =
M = l
u = —A>
x — y = l x-y = -4
Vdi: x - y = ket hdp vdi (2) ta diTdc he:
x — y = l
2x + 3y=^\2^'
Vdi: x - y = -4 k^'t hdp vd,i (2) ta difdc he:
x-y = -4 2x + 3y = \2
x = y = x = y =
Vay he da cho cd hai nghiem: [x = 3;y = 2);(x = 0;y = 4)
JfB^ 6.33: Tim tat ca cac cap so' (x, y) thoa phiTdng trinh:
5x-2^f^ [2 +y) + y^ +1 = 0^
I
(88)Taco; 5x-2yfx {2 +y) + + \
<^ (^4x-4^ + l^ + (^y^ - lyyfx +x^ = < ^ ( V I- l) ' + ( > ' - ^ / I ) ' =
(l^-lf =0 ^
[y-^f=0
2yfx-\ y-y[^ = 27^ =
Vay cap s6' (x, y) phai tim la
'• =
2-J_ 1^ 4'2
6.34: Giai he phi/dng trinh: - x + y =
- 5A: + =
A:-V)^ + = x^-5x + y=:Q (1) (2)
• He phiTdng trinh xac dinh vdi A: +1 > ;c > - (*) Khidd: = ^ + <^>' = ;c'+2A; + (3) Thay vao (1) ta diTdc: 2A;^-3x + l = 1 thoa (*) Thay vao (3) c6 y^ =4,y.=-
4
Vay nghiem cua he phu-dng trinh la: [x = l;y = 4); 1 91
174
PC' 6.35: Giai he phtfdng trinh:
J_ _
X y — 2
X 2-y
- = Gi§i
Dilu kidn: x^O y-2^0' Dat:
1 M = — X V = 2-y thihe ^ M + 3v = 2M^V =
5 7
7 5 < ^
• 7
7 3
7 1
Vay he da cho c6 mot nghiem 7
6.36: Giai he phtfdng trinh: +y^+xy^l
/ + / + ; c V = x^+y^+xy=^l
x' - f / + ; c ' > ' ' = x^ +y' = l-xy
{l-xyf-xV =21
Dieu kien: xva y cung dau Khi do: (1)
4^
x'^ + y^ = l-xy [x'+/f-2x'y'+x'/=2l
'x^+y^=l- xy
49_14;cy = 21 xxy = '+y'=5
x'+/=5
xy=4
vay x\y^ la nghiamcua phuTdng trinh : t^-5t + = t = l
t =
(1)
(89)Do dd hoac
V i X va y cOng dau nen ta c6:
x'=4 ;c = ± l
2 ^ hoac
y = ±2 •
x = ±2
x = l y = hoac
x = -\ x = y = -2' hoac y = -2' ^ = •
x = -2 y = - i Vay nghiem cua he phu^dng trinh da cho Yk:
x = l x = -l 'x = x = -2 y = 2' y = -2' y = i'' y = - i
f>€ 6.37: Giai vk bien luan theo tham so a he phiTdng trinh:
X -xy + ay = (1)
y -xy~4ax = (2)
Na'u a = 0,.he trd thanh: x^-xy =
y^ -xy = • Neu a^O:
x{x-y) = y{y-x) =
T a c d : ( l ) ^ ; c ^ = ; ^ ( ; c - a ) ^ ; =
x=0 V x=y y=0 V x=y
(vdi x^a) ^ x = y
x-a Tha'y v a o ( ) ta cd:
x-a x-a •-4ax =
•a)
-x^ [x-a)-4ax[x-af = ^ ax^-4ax(^x^ -2ax + a^) =
^ax[-3x^ + Sax-4a^) =
<r^x = 0 V 3;c^-8ajc + 4a^ = (via^O) ^x = 0 V x = 2a V A: = — ( t h o a d i l u k i e n
3
Vay ta cd: * Neu:a = thi x = y (vdi xeR)lk nghiem 176
* N e u : a ^ O t h i nghiem gdm:
[x^y^0);{x = 2a,y = 4a); la Aa\
6.38: T i m m de he phu'dng trinh sau v6 nghiem:
X + Imy = (1) 2mx-6my = 4m + (2)
Gi§i
* K h i m = 0: thi (2) v6 nghiem nen he v6 nghiem * K h i m ^ :
T i i r ( l ) t a c d : x + 6my = (!') C6ng ve theo ve (1') va (2) thi cd:
{2m + 3)x = 4m + (3)
• Neu 2OT + ^ (tiJcla m ^ - - ) t h i (3)<^ x = ^ ^ ^ ^ ^ t ^ = 2 m + ( l ) ^ y = l-x
2m (vdi m^O) Luc dd he cd nghiem Neu m = -^ thi (3) ^Qx = (v6 dinh)
1 — X
Con y = = •—(1 - x) n^n he cd v6 s6'nghiem 2m
T d m l a i he phiTcIng trinh v6 nghiem m = 6.39: Giai phu'dng trinh: ;c' + = 2lJ2x-\
Phufdng trinh duTdc viet thanh: x^ = 2^2x-l - Dat t = U2x-l^t' = 2x-\
Do dd phiTdng trinh trd he phu'dng trinh sau:
x' = f - l (1) ^ = 2x-\)
Trilf ve theo \€ (1) cho (2) thi:
(90)<=^t = x V x"- + tx + t^ + 2^0
^ t = x 'vix^ + tx + t^ +2 = c6 A = t^-4[t^+2) <0:vdnghiem) The : t = x vao (1) ta difcJc:
x' -2x + \ 0^{x- -l)(x'+x- - ) =
f (1)
6.40: G i a i he phifdng trinh: 2x' ' = l + x^
[ + / (2)
(3) •
Gidi
V d i X = 0 thi (1) =^ z = va v d i X = 0 thi (2) cho y = Do dd X = y = z = la mot nghiem cua he
Bay gid ta x ^ t x>0,y>0,z>0:
(Lm y rang (1), (2), (3) cho ta ^ > 0,^ > 0,z > )
Ta biet rang l + a^>2a<=^ < va dau " = " chi xay
1 + a a ^ = l
Do dd t a c d : (\)=^x = z.- 2z l + z' i2)=^y = X
(3)^z = y.-2x \ x
2y 2 -<x
•<y
1 + /
Suy ra: y<x<z<y nen phai cd:
y = X = z \uc &y y^ = x^ = z^ = \ y = X = z = K e t luan: N g h i e m cua he gom: {x = y = z = 0);{x = y = z = l) •
178
CHl/OfNG VII: DO THI CUA HAM SO
De 7.1: V e thi cua ham so": y = yjx^ -4x + Ta cd: y = V - ^ ^ -4x + = ^(x - ) ' = |x -
x — 2 neu x>2 2 — x neu x <2 • M X D : R
• Bang gia t r i :
• V e :
x 0 2 4
1
y 2 0 2
Nhan xet: D o thi cua ham so la hai tia A B va A C v d i A ( , 0), B(0, 2), C(4, 2)
De 7.2: Cho ham so y = f(x) = 2-yjx'' -2x + \
1) V e thi cua ham so' tren
2) T i m tat ca cac gia tri cua x cho f(x) < G i i i
1) T a c d : y = f(x) = 2-yjx^-2x + \ = 2-yj(x-\) = - | x - l
(91)2-x + l neu x>l
2 + X — neu x <l T X D : R
Bang gia tri:
3 — X neu x>l
x + \u x<\
V e :
X
y
• Nhan xet: Do thi ham so f(x) la hai tia A B , AC vdi A ( l ; ) , B(0;1),C(2;1)
2) / ( x ) < l < ^ - x - l <1<=^ x - >
x-l>l
j c - K -
x>2 x<0
Vay X > hoat jc < thi f(x) < I
De 7.3: Cho ham so: y = yf]^ + ^Jx^ -4x +
1) T i m tap xac dinh cua ham so'
2) Rut gon y (loai bo dau V~ va dau 11) 3) Ve thi ham so
4) T i m gia tri nho nhaft cua y ya cac gia tri tifdng iJng cua x 5) TCr thi hay chi true do'i xiJng cua thi da ve d cau va
dung phep toan de ch^ng minh dieu
1) Ham so" xac dinh <^
G i i i
x^>0
x^ - 4A: + >
x^>Q
{x~2f>Q ^ XER
180
Vay : T X D c i l a y Ik R
2) y = ^ + ^{x-2f = | ; C | + | A: -
x + x-2 neu x>2
= • x-x + 2 neu < ;c < =
- x - x + 2 neu j c < 3) T X D : R
• Bang gii tri: ,j
2 x - n6u x>2
2 n€u 0<x<2 -2x + neu x<0
Do thi ham so' la hai tia A C , BD va doan t h i n g A B v d i A(0; 2), B ( ; ) , C ( - ; ) , D ( ; )
4) Nhin vao thi ta tha'y gia tri nho nha't cua y la 2, dat duTdc <l=!>0<;c<2
5) • TO thi nhan true do'i xitng ciia d6 thi la x = • Goi M (.«o!3'o) 1^ diem thuoc thi
Goi M'{x',;y',) la diem doi x i f n g c i i a M qua durdng t h i n g =
K h i d6: yo = / o v« ^ x', =2-x, ^ =2-x',
=^ y0 = >o = 1^01 +1^0 - h |2 - xo' 1 +1(2 - ) -
(92)M' thu6c thi ham so
Vay thi ham so c6 true doi xiJng la x =
De 7.4: Ve thi cua ham so: >' = x- \ \
Ta c6: y =
Giii
- - l | neu x > 1 - J: - l | neu x <
x-2 neu x>2 2-x neu l<x<2
X neu < ;c <
-X neu X <0
x-2 neu x > neu A: <
TXD: R Bang gia tri:
Ve:
X - 1 2 3
y 1
NhSn xet; Do thi cua ham so la hai tia OC, BD va hai doan OA, AB vdi A ( l ; 1), B(2; 0), C ( - ; 1), D(3; 1)
182
Dt' 7.5:
1) Ve thi cua cac ham so' sau day tren ciing mot he true toa do: y^x^-l (1) y = -x^-2x + 3 (2)
2) Chitng minh giao diem cua hai thi n6i tr6n luon thuoc thi ciia ham so':
1 y =
k + i {l-k)x^-2kx + 3k-l ydik^±l
Gi^i
1) Ve (P,):y = - • TXD: R • Bang gia tri:
X -2 - 1
y = x'-\ 3 -
• Do thi la mot parabol c6 dinh (0; -1) nhan true tung lam tnic do'i xiJng cat true hoanh tai cac diem (1; 0), (-1; 0) • Ve {P^):y = -x'-2x +
• TXD: R • Bang gia tri:
- - - 1
y = -x^-2x + 3
• Do thi la mot parabol c6 dinh ( - 1; 4) nhan dtfdng thing x = - lam true do'i xiJng
B6 thi cat true hoanh tai cac diem (1; 0), ( - 3; 0)
(93)2) Hoanh giao diem cua (fj) va [P^) la nghiem ci5a phU"dng trinh: -1 = -x" -Q.x + 2;c^ + 2;c - =
<^ = l , x = - n6n cac giao diem la A ( l ; 0), B ( - 2; 3)
* X ^ t dd thi (C) cua h^m %6: y = " ^ ^ [ ( l - ^ ) ^ ' - 2/:x + 3ik - Taco A G ( C ) < ^ = l - i t - y t + ) t - l Dieu lu6n diing
5 € (C) ^ = - ^ [ ( - i t ) + / : + i t -
<!=> 3it + = - 4/: + 4)t + 3/: -
3)t + = 3A: + Dieu luon dung Vay cac giao diem A va B luon thuoc thi ( Q s j
7.6: Trong cung he true vuong gdc cho parabol ( P ) : j = — va du'dng thdng (D) qua diem / cd he s6'gde m
1) Ve (P) va vie't phu-dng trinh cua (D) 2) Tim m cho (D) ti6p xiic vdi (P)
3) Tim m cho (D) va (P)cd hai diem chung phan biet
1) • T X D : R
m
• Bang gia tri:
X - - 2 4
x''
' = 1
Ve:
184
-4 -3 -2 -1 q
• Nhan x^t: Dd thi ham s6' y = ^ la mot parabol cd dinh
0(0; 0) la diem ctfc tieu, nam phia tren tnie hoanh ; tnic tung la true do'i xiJng
• Dqdng thing (D) cd he so' gdc bang m
nen phu'dng trinh d^dng thang (D) cd dang: y = mx + b
3 Theo gia thie't: I e (D) ^ - I = -m + b =^ b ^ — m -
^ ^ 2
Vay phu'dng trinh du'cJng thang (D) \k y = mx — —m — l • 2) Phu'dng trinh hoanh d6 giao diem cua (D) va (P):
x^
— = mx m-l 4^ x^-4inx + 6m + = (*) 4
Ta ed: A ' = 4/7ji - 6m - (D) tiep xiic (P) ^A' = 0^
m = m — — 3) (D) va (P) ed hai diem ehung phan biet
<^ (*) cd hai nghiem phan biet • <^ A ' > < i ^ m ^ - m - >
m > m < - -
(94)De 7.7: Trong ciing he true vuong gdc, cho parabol (P): y = - — x^ va du'dng thang (D).: y = mx - 2m - 4
1) Ve(P)
2) Tim m cho (D) tiep xilc vdi (P)
3) ChiJng to (D) luon luon qua diem co dinh A e (P) GiSi
1) • TXD:R • Bang gia tri:
X - - 0 2
4
' - - - 0 - -
Ve:
' • Nhan x6v Do thi ham so' j = - ^ x ^ la mot parabol cd dinh 0(0; 0) la diem ciTc dai, nam phia du'di true hoanh ; triic tung la triic do'i xiJng
2) Phyong trinh hoanh giao diem cua (D) (P) x^
=^mx — 2m — \<^x^— Amx — 8m — = 4
A' = 4m' +8m + = 4(m + l)' (D) ti6p xiic (P) <^ A' = 0 4=^ (m + if =
<^m + l = 0<^/n = - l ?
3) Goi ) la diem co' dinh thuoc (D)
^0 — '^^0 — 2m — diing vdi mpi m m(xo - ) - l - = Odung vdi moi m
x,-2 = x,=2
[-l-y^=0 [y,=-l Vay (D) lu6n qua di^m c6'dinh A(2; -1)
-2^
Matkhac: A e (P) - = — ^ - = - (dung) Vay A e (P) Tdm lai: (D) luon luon qua diem co'dinh A e (P)
Oi 7.8: Trong mat phing toa cho parabol (P) -.y = va diTdng thing (D): y = x +
1) Khaosatvaveddthi(P)cuahams6': >' = x \ 2) Ve (D)
3) Tim toa giao di^m A va B cua (P) va (D) bl[ng 66 thi va phep toan
4) Tur A va B ve AH ± Ox ; BK lOx Tinh dien tich ciia ti? giac AHKB
II) Khaosdtva ve (P): y = x\ • TXD: R
• Vi a = > n6n ham s6' nghich bien khoSng x < va dong bien khoang x >
• Bang gia tri:
X -2 - 0 1 2
y = x' 1 0 1 4
• Diem cifc ti^u la 0(0; 0) (do a = > 0)
E)6 thi la mot parabol c6 dinh O (diem cufc tieu) n^m phia tren true hoanh va c6 true do'i xitng la Oy
(95)2) (D): y = X + la diTdng thing qua hai diem C(x = ; y = 2) va E ( x = - ; y = 0)
3) * Bang thi: Can ciJ vao thi cua (P) va (D) hai giao diem A va B CO toa la: A ( - 1; 1), B(2; 4)
* Bang phep toan: Toa giao diem cua (D) va (P) la nghiem cua he phu'dng trinh:
'y = x' (P) y = x + (D) Ta suy phqcfng trinh hoanh giao diem la:
=x + 2^ x^ -x-2 =
Phu'dng trinh c6 a - ^ + c = 0(l - ( - ) - = O), nen c6 hai nghiem la: ;c =—1 ; x — l
Thay x = - vao (D) hoac (P) ta dqdc: >- = - + = Thay x = 2, ta du'dc y =
Vay nghiem cua he phu'dng trinh tren la: ( x = - ;y = l).VayA(-l; 1)
(X = ; y = 4) Vay B( ; )
4) Theohinhve: AHKB la hinh thang vuong c6 hai day la AH, BK,dirdngcaolaHK
Dt{AHKB) = ^{AH + BK)HK
• Dya vao thi ta c6:
188
BK = \
1 =
HK^HO + OK (vl O nam giffa H, K)
Vay: D<(A//0) =-^-(l + 4).3 = y (dvdt)
+ \x, = + = p6 7.9: Cho (P) la thi cua ham s6' y = ax^ va diem A ( - ; - 1)
trong cdng he true
1) Tim a cho A € (P); ve (P) (vdi a viTa tim difdc)
2) Gpi B e (P) c6 hoanh la 4; vie't phu'dng trinh du-dng thing
AB
3) Viet phu'dng trinh dufdng thing tiep xiic vdi (P) wk song song : vdi AB
1 1) Tacd: A(-2;l)G(P)<^-l = a-(-2) <^« = -
Vay iPy,y = ~ x \ • TXD: R
• Bang gia tri: X - - 2 4
1 - - - -
• Ve: y /
-4 -3 -2 -1 1
' / -4
(96)• Nhan x6t: Do thi ham so y = ~^ la mot parabol c6 dini, 0 ( ; 0) Ik diem eye dai, nam phia difdi true hoanh, true tung la true do'i xiJng
2) B{x,;y,)e(P)^y,=-^xl^y,=-4.y'^yB(4;-4) Phu-dng trinh dudng t h i n g A B eo dang y = ax + b (vi A B kh6ng
song song v d i Oy) Ta eo: AeAB
BEAB
-l = -2a + b -4 = 4a + b
6a = - b = -\ 2a 2
b = -l+2
\_
2 b^-2
Vay: Phu-dng trinh dudng t h i n g A B la : j = ~ x -
3) Phtfcfng trinh du'dng t h i n g ean t i m c6 dang y = a'x-\-b' (d) {d)IIAB:^a' = ~-b'^~2^{d):y = -]^x + b' Phufdng trinh hoanh giao diem eiia (d) va A B :
x^
—- = x + b' ^ x^ -2x + 4b' = {) 4
A ' = l - ^ '
(d) tiep xuc (P) A ' = - ^ ' = <^ 4Z?' = <^ Z?' = i PhiTdng trinh du'dng t h i n g ean t i m la : = - i A; +
-2
—"1
7.10: Cho parabol (P) : y = jx^ va du'dng t h i n g (D) qua hai d i e m A, B tren (P) c6 hoanh Ian lUdt la: - va
1) K h i o sit sir bien thien va ve thi (P) eua ham so'tren
190
2) 3)
V i e t phUdng trinh du'dng thang (D)
T i m diem M tren eung A B cua (?) (tufdng \ing hoanh
X g f - ; ] ) eho tam giae M A B eo dien tieh Idn nhaft
Gi§i 1) T X D : R
1 • SU bien thien: ham so eo dang y = ax v d i a = — > nen ham so nghich bien x < 0, dong bien x > 0, bang x =
• Bang gia t r i :
x - - 2 x'
4 1 Ve:
• Nhan xet: Do thi ham so >' = — la mot parabol c6 dinh
0 ( ; 0) la diem eUe tieu, nam phia treri true hoanh, true tung la true doi xiJng
2) PhiTdng trinh dudng t h i n g AB CO dang y = ax + b (vi durdng ^ thiniT A B khong song song vdi Oy)
(97)Be{P)=^y, = ^ = Vay B(4; 4) Ta cd: AeAB BeAB l^-2a + b 4 = 4a + b
1 a = — 2
-2.- + b = l 2
6a = -2a + b = l
1 a = — b = Phifcfng trinh difdng thang AB y = +
3) PhiTdng trinh diTdng thang (d) song song AB cd dang y^^x + b
Phifdng trinh hoanh dp giao diem cua (d) va (P) la: Lx^ =-x + b^ -2x-4b = 4
2-A' = l + 4b (*)
(d) tia'p xuc(?) ^A' = ^ l + 4b = 0^b = - - 4 Khi: b = - - nghi^m kdp cua (*) la x = 4
Khi dd y = - l ' M 4 la diem can tim That vay moi di^m M' khac M thupc cung AB ciia (P) deu cd khoang each tu" M' den AB nho hPn khoang each tOf M den AB (vi M' nim giffa hai dirdng thang song song d va AB)
Diem M can tim la M
De 7.11: Cho ham so y = ax^ cd dd thi (P)
1) Tim a biet rang (P) qua diem A(l; - 1) Ve (P) vdi a vffa tim dffPc
2) Tren (P) lay diem B cd hoanh dp - 2, tim phffdng trinh cua
192
dUdng thang AB va tim tpa dp giao diem D cua dudng thang AB va true tung
3) Vie't phu'dng trinh dufdng thang (d) qua O va song song vdi AB, xac dinh tpa dp giao diem C cua (d) va (P) (C khac O) 4) Chu'ng to OCDA la hinh vuong
1) * (P): y = ax^
A(l;-1) e(P)^>'^ ^ax\\ a{\f <^a = - l Vay phiTdng trinh cua (P) Va y = -x^ * Ve(P):
* TXD: R
X -2 - 0 1 2
y = -x^ - - 0 - - (P) la parabol cd dinh 0(0; 0) (diem cUc dai vi a = - < 0), nam phia dirdi true hoanh ; true doi xiJng la true tung Oy
'2) Ta cd: B G (P) ^y,= -x\ - ( - ) ' = - VSy B( -2; -4) * Phu'dng trinh du-dng thing AB ed dang y = ax + b
(98)TOd6,ta CO he: b = -\ a - a = - +
a + Z? = - l \b = ~\-a
-la + b = -A [ - a + ( - l - a ) = - a = l
Z7 = - l - l = -
Vay phiTcfng trinh cua dicing thang A B la: y = x -
Giao d i ^ m D ciia dtfdng thang A B va true tung c6 hoanh = O v a = ^ - = - = - Vay: D ( ; - ) 3) X d t du-dng thang (d) qua O c6 dang : y = a'x
• {d)ll[AB) ^a = a' (hai he so'gdc bang nhau)
V d i a = 1 (he goc cua (AB) : y = x - 2) nen (d) c6 he so gu,
a' = Vay (d) : y = x
• Toa giao diem C cua (P) va (d) la nghiem cua h ^ phu-dng
trinh:
y = x [y = -x\
Suy ra: -x^ = x<i=4>x^+jr = 0<^A: = 0;jc = -
X = la hoanh giao diem O
X = - 1 la ho^nh giao diem C Suy ra: tung cua C la : = jc^ = -
V a y : C ( - l ; - l ) 4) O C D A la hinh vuong
= >'c = - 1 (A va C cung tung do) n6n dtfdng thang A( song song vdi true Ox, suy ra: AC _L OD tai I
• I la trung diem cua doan A C vi cd:
IA=x, = • i =
IC= - = I cung la trung diem OD vi cd:
10 = - =
ID = OD- 01 (I nam giffa O va D)
1 | y, - 2 — 1 = - =
Vay: Tu" giac O A D C cd hai dudng cheo cat tai trung diem chung I , vuong gdc vdi va bang (AC = OD = 2) nen la hinh vuong
7.12: Trong cung mat phang toa cho hai du'dng thang:
{D,):y = x + \ (D^): x + 2^ + =
1 ) T i m toa giao diem A cua ( D , ) va ( D ^ ) bang thi va kiem tra lai bang phep toan
2) T i m a ham so' y = ax^ cd thi (P) qua A ; khao sat va ve thi (P)
3) Tim phUdng trinh cua du'dng thang tiep xuc vdi (P) tai A Giii
1) • Bang thi:
Xet hai difdng thang: ( D , ) : = X + 1 va ( D 2) : X + 2y + = hay: y = — — x —
• Bang gia tri:
X
y = x + \ 1
1
- - Ve:
(99)(Z),) la audng thang qua (0; 1) va (2; 3) {D^) la du-dng thang qua (0; - 2) va (2; - 3)
Nhin vao thi ta thay {D^ ) cat (D^ ) tai A ( - 2; - 1) • Bang phep toan:
PhiTdng trinh hoanh giao diem cua (DO va ( D ) : X + = X -2 <^ x = - (y = - l ) Vay (Z),) cat ( D j tai A ( - ; - )
2) (Py y = ax'qua A^-l = a.{-2f ^a = ~^ Way: {P):y = -^x\
• T X D : R
• S\i bien thien: ham so c6 dangy = ax^ c6 a = - - <
4
nen : Ham so dong bien x < 0, nghich bien x > 0, bang khi X =
• Bang gia tri:
X - - 2
- - - - • Ve:
• Nhan xet: Do thi ham so' y = la mot parabol c6
dinh 0(0; 0) la diem cu'c dai, nam phia difdi true hoanh, true tung la true do'i xiJng
3) Du'dng thang (D) khong song song Oy nen phtfcfng trinh eo dang y = ax + b
Ae{D)^-\ a.[-2) + b^b = 2a-\ Vay: [D): y = ax+ 2a-l
Phu-dng trinh hoanh giao diem cua (D) va (P) x'^ =ax + 2a-\^ x^+4ax + Sa-4 =
4
A ' = a ^ - a + = ( a - l ) ^
(D) tiep xiie (P) A ' = 4^ (fl - ) ' =
< ^ a - l = < ^ a = l
a = l taco b = 2.1 = -Vay: phu-dng trinh du-dng thang (D) la : y = x +
i 7.13: Trong eung he true toa goi (P) la thi eiia ham so' '•y = ax^ va (D) la thi ciia ham so' y = - x + m
1) T i m a biet rang (P) qua A(2, -1) va ve (P) vdi a t i m du-de 2) T i m m cho (D) tie'p xiic vdi (P) (d eau 1) va t i m toa
tiep diem
3) Goi B la giao diem cua (D) (d eau 2) vdi true tung; C la diem do'i xiJng cua A qua true tung Chitng to C nam tren (P) va tam giac ABC vuong can
Gi^i
A(2; -l)e{P)^ y^^axl^-\ a.2^ ^a = -^ Vay {P):y = ~ x '
• T X D : R • Bang gia t r i :
197
(100)X - - 0 2 4
- - 0 - - Ve (P):
Nhan xet: Do thi ham so y^ x^ la mot Parabol c6 4
dinh 0(0; 0) la diem ciTc dai, nam phia diTdi true hoanh, true tung la true do'i xiJng
2) PhUdng trinh hoanh giao diem ciia (D) va (P):
= - x + m<^jr - x + 4«i = 4
A' = - m
(D) tiep xiie (P) A' = ^ - 4m = ^ 4m = ^ m = Khi nghiem kep eua phu"cfng trinh la x =
Vdi x = 2thi y = (2f = - 4
Toa tiep diem la A(2; - 1)
3) C doi xiJng eua A(2; - 1) qua true tung nen C(-2; - 1) • Gia siiCe{P)<=>y,=axl^-\ -^{'^f ^ - = -
(diing) Vay Ce(P) Ta eo AC cat true tung tai M(0;-1)
Vi BM = MC = MA = AA5C vuong tai B 198
Ma EC = BA (Do A, C d6'i xitng qua true tung va B thuoc true tung)
Do d6 tarn giac ABC vuong can tai B
i 7.14: Cho thi {C):y = x'^-2mx~4 va diTdng thang (D): y = 2x
1) ChiJng minh (D) luon cat (C) tai hai diem phan biet A va B 2) Tim M de khoang each AB la nho nhat Gi^i
1) Hoanh giao diem giffa (C) va (D) la nghiem eila phtfdng [ trinh:
x'-2mx-4 = 2x^x'-2(m + l)x-4 = (*)
Taeo A' = (m + l ) ' + > ;Vm
Vay (D) luon c^t (C) tai hai diem phan biet A va B 2) Dat A{x,;y,),B{x,;y,) thic6:
= 2x, ; y, = 2x, nen:
AB^ = {x^ x,f + (2x, 2x, f =5 (x^ x, f
-=
ma (*) cho ta: x^+x^=2{m + \),x^x^ = - nen AB' ^5 4(m + l f + l ] = 2o[(m + l f + Do vay doan AB ngin nhat k h i m + l = < ^ m = - l
DC 7.15: Trong he toa vuong goc Oxy cho ba diem A(2; 5), B ( - l ; - l ) v a C ( ; )
1) Chu-ng minh ba diem A, B, C th^ng hang
2) Chifng minh du-dng thing AB va cac difdng thing (^i ),(^2)
c6 phtfdng trinh la y = 3, = -^{x -1) la ddng quy
Gi§i
1) Xet du-dng thang AB cd phu-dng trinh dang y = ax + b
(101)(1) (2) Dirdng thing qua A nen: 2a + b =
Di/cing t h i n g qua B nen: — a + b = —
Giai hd (1) va (2) ta c6 : a = 2, b = ,„ Do 66 d\Xdng thing A B c6 phu-dng trinh la : y = 2x +
Ta c6 : = 2.4 + = nen diem C thuoc diTdng t h i n g A j Vay ba diem A, B, C t h i n g hang
2) To a giao cua [d^) va du'dng thang A B thoa :
y = y = 3x + l
'.^ =
y =
Ta l a i cd = - ( - 7) nen giao di^m ay thuoc ( j , ) Vay cac dtfdng thing A B , {d,),{d,) ddng quy
7.16: Cho thi (C) cua ham so' y = ax^ +bx + c
1) Dinh a, b, c de (C) d i qua ba diem sau: A(0 ; - ) ;B(1 ; - - ) ; C ( - ; )
2) T i m phufdng trinh du-cJng t h i n g di qua M ( ; - 8) va tiep xiic (C) 3) T i m toa tiep diem cau
1) Do thi (C) di qua A , B, C nen ta c6:
c = -
a + b + c = -6 <^ 9a-3b + c = 14
a + b = ~2
9a-3b = \S<^a = l,b = -3,Q^-4
c = -
Vay a = l , b = - 3,c = -
2) Du-dng t h i n g (d) qua M(0, -8) c6 phu-dng trinh dang: y = mx - Phu-dng trinh hoanh giao diem giffa (d.) va (C) la:
x^ -3x-4 = mx-S^ x^ -{3 + m)x + = (*)
De (d) tia'p xuc (C) thi ph^i c6 A = <^ {3 + mf - \6 = ^ ( m + ) ( m - l ) =
4 = >/ n = l , m = —
200
Vay cd hai dtfdng thing (d) can tim vdi phu'dng trinh la:
{d,):y = x-S [d,):y = -lx-%
3) * V d i (J;) thi (*) tra thanh: ;c^-4jc + = 0<i=>jc = nen tiep diem cd toa la : (2, — 6)
* V d i [d^) thi (*) trd thanh: x^ -^4x + = Q ^ x = -2 nen tiep diem cd toa la : ( — 2; 6)
De 7.17: Cho hai Parabol [P^):y = x^
{P,):y = x'-2x-\
Vie't phu'dng trinh du-dng t h i n g (d) tiep xiic v d i ca {P^) va (P^) •
Giii
• Du'dng t h i n g (d) cd phu'dng trinh dang : y = ax + b • Phu'dng trinh hoanh giao diem giiJa (d) va (Pj) la:
x^ =^ax-\-b x^ -ax-b =
Dieu kien d^ (d) tiep xuc vdi (P^) la Ai =
^a'^4b = 0 (1)
• Phu'dng trinh hoanh giao diem giiJa (d) va [P^) la:
A: ' - 2x - = ax + ^ ^ - (2 + a);c - (1 +Z J) =
Dieu kien de (d) tiep xuc vdi (P^) la A 2 = ^ ( + a ) ' + ( l + Z7) =
+ a + 4Z? + = (2) Tuf (1), ta cd: 4b = - a ' The 4b vao (2) ta du'dc:
(102)CHlTdNG V I I I : MOT SO BAI TOAN
V E TO H0P
P H E P E ) E M - D O N G T i /
D e : H a i dia d i e m A , B each 60km Ngifdi d i xe dap khcii hanh tu" A den B r d i tiT B trci A v d i van toe nhif luc dau, nhifng sau k h i d i tu* B du'dc g i d thi nghi 20 phut r d i d i t i e p ve A v d i van toe tang t h e m 4km/h T h d i gian d i va ve bang T i n h van toe ban dau
Gi^i
G o i X km/h la van toe ban dau ciia ngtfdi d i xe dap (dieu k i e n : x > 0)
T h d i gian d i tu: A den B : — g i d
X
T a c : p h u t = i gicf
T h d i gian d i tu" B ve A : 1 + i +
3 x^A J
gid , hay : ;4 ^ - ; c '
3 ;c + , gid
Theo dau bai ta c6 phu-dng trinh:
60 , 60-x x = 2Q
X = - ( l o a i ) Vay van toe liie ban dau: 20km/h
D e 8.2: M o t ngu-cfi d i oto tir A tdi B v d i van toe 30km/h Sau mot thdi gian mot ngu-di d i moto v d i van t6c 40km/h va neu gii? nguyen _van toe nhtf the^ thi se b^t kip olo d t a i B The^ nhtfng k h i d i dtfde nij-a 202!
quang du"dng A B thi moto tang van toe len 45km/h va sau gid da bat kip oto T i n h quang dtfdng A B
Gi§i
Dat quang du'dng A B = x (x > 0)
T h d i gian ma oto ehay h e l quang du'dng la: = 30 T h d i gian ma moto ehay het quang du'dng v d i van toe k m / h l a : L =
40;
De ea hai gap hie vtfa t d i B thi thdi gian moto ehay sau oto la: L = — - — =
0 2 Ar\ 30 40 120
TiJ* lue k h d i hanh luc gap moto can thdi gian la:
40 80
NiJa quang du'dng dau, moto ehay v d i van toe 40km/h va quang du'dng moto da d i du'de la: — +
Til Mc k h d i hanh t d i luc gap oto can t h d i gian la: - + 45 „
2 = ^ +
30 60
X
V i moto k h d i hanh sau oto thdi gian la: = nen ta c6:
_x_ _
60 ~ 80 • +
120
X X , ^ X , ^ X
+ ^ - + - - + 10 + — , 120 12
A: x X
i 12 = < ^ x = 120
Vay quang du'dng A B dai 120km
(103)Dd 8.3: Mot r^hom hoc sinh diTdc giao nhiem vu trdng 80 cay thong NhiTng thuTc hien nhdm ay diTdc tang cifdng them ban, d6 moi ban da trdng it hdn cay so vdi du" dinh Hoi hic dau nhom
CO bao nhieu hoc sinh ? (Biet rang so' cay moi ban trdng nhuf nhau)
Gi^i
Goi X la s6'hoc sinh c6 lilc dau cua nh6m (x nguyen du'dng) 80
Luc dau moi ban du'dc giao trdng — cay X
Nh^ng thi/c hien c6 them ban nffa so'hoc sinh tham gia 80
trdng la (x + 4) ban, do moi ban trdng du'dc cay x +
Theo dau bai cd phifdng trinh:
- - — = l h a y x ^ + x - = X x +
Giai diWc x = 16, thda man dieu kien bai toan Vay sd'hoc sinh tham gia trdng cay liic dau la 16 em
Dd 8.4: Cho tam giac ABC vudng tai A c6 chu vi 12m va tong binh phu-dng cua ba canh la 50 Tinh dai ba canh cua tam giac ABC
Giii
Dat: AB = x (m), AC = y (m), BC = z (m) Giai su": x <y <z
TJieo dau bai ta c6 he phrfdng trinh:
X + y + x = l2 (1)
x^+y^+1^=50 (2)
Theo dinh ly Pitago, trong AABC : x^ =^z^ Thay + y^ = vao phifdng trinh (2) :
204
2z^ = 50 <^ = 25 <^ x= Z = - 5 (loai) D o d : x ' + / = (3)
Thay z = v a o ( l / x + y = (4)
Tiir(3) va (4) ta c6 he: x + y = l x'+y'=25
x + y = l
{x + yf -2xy = 25
5 =
S- -2P = 25 P = 12 Vay X , y la nghiem cua phu'dng trinh bac hai sau:
fx =
Vay:
-1X + 12 = 0^ x = \x =
V y = [y =
X =
•Vi: X <y<z nen chon x = y = Vay canh AB = 3m, canh AC = 4m va canh huyen BC = 5m
De 8.5: Mot ngu'di mang binh lit di mita lit sffa Chu quan lai chi
CO mot binh 12 lit day sffa va mot binh khdng Slit Lam the' nao de dong dffdc lit sffa vao binh lit ?
Binh 12 lit Binh lit Binh lit
Lan
Lan S
• Lan
Lan
1
(104)D e 8.6: Chia mot hinh vuong cd canh bang 10^/2 cm l a 100 hinh vuong nho bang (xem hinh v e ) X e p 201 d i e m vao ben hinh vuong (cac d i e m deu d ben cac hinh vuong nho)
-D e 8.6: Chia mot hinh vuong cd canh bang 10^/2 cm l a 100 hinh vuong nho bang (xem hinh v e ) X e p 201 d i e m vao ben hinh vuong (cac d i e m deu d ben cac hinh vuong nho) D e 8.6: Chia mot hinh vuong cd canh bang 10^/2 cm l a 100 hinh vuong nho bang (xem hinh v e ) X e p 201 d i e m vao ben hinh vuong (cac d i e m deu d ben cac hinh vuong nho) D e 8.6: Chia mot hinh vuong cd canh bang 10^/2 cm l a 100 hinh vuong nho bang (xem hinh v e ) X e p 201 d i e m vao ben hinh vuong (cac d i e m deu d ben cac hinh vuong nho) D e 8.6: Chia mot hinh vuong cd canh bang 10^/2 cm l a 100 hinh vuong nho bang (xem hinh v e ) X e p 201 d i e m vao ben hinh vuong (cac d i e m deu d ben cac hinh vuong nho) D e 8.6: Chia mot hinh vuong cd canh bang 10^/2 cm l a 100 hinh vuong nho bang (xem hinh v e ) X e p 201 d i e m vao ben hinh vuong (cac d i e m deu d ben cac hinh vuong nho) D e 8.6: Chia mot hinh vuong cd canh bang 10^/2 cm l a 100 hinh vuong nho bang (xem hinh v e ) X e p 201 d i e m vao ben hinh vuong (cac d i e m deu d ben cac hinh vuong nho) D e 8.6: Chia mot hinh vuong cd canh bang 10^/2 cm l a 100 hinh vuong nho bang (xem hinh v e ) X e p 201 d i e m vao ben hinh vuong (cac d i e m deu d ben cac hinh vuong nho) D e 8.6: Chia mot hinh vuong cd canh bang 10^/2 cm l a 100 hinh vuong nho bang (xem hinh v e ) X e p 201 d i e m vao ben hinh vuong (cac d i e m deu d ben cac hinh vuong nho) D e 8.6: Chia mot hinh vuong cd canh bang 10^/2 cm l a 100 hinh vuong nho bang (xem hinh v e ) X e p 201 d i e m vao ben hinh vuong (cac d i e m deu d ben cac hinh vuong nho)
u-iiung iiiinu l a i i g iiic K C uuyu i i i y i uuung
tron CO ban kinh 1cm chita it nha't ba d i e m 201 d i e m n d i tren
Giai
Phan pho'i 201 d i e m vao 100 hinh vuong nho thi c6 it nhat mot hinh vuong nho chiJa it nhat ba d i e m (Nguyen ly Dirichlet)
Ta CO canh hinh vuong Idn la > ^ c m thi canh hinh vu6ng nho la ^/2 cm, suy ban kinh du'dng
tron ngoai tie'p hinh vuong la 1cm Du'dng tron chiJa hinh vuong nho nen chiJa ca ba d i e m ben hinh vuong nho ay V a y ke dtfdc du'dng tron ban kinh 1cm chtta it nhat ba d i e m
201 d i e m da cho
D e 8.7: Chia mot hinh vuong hang va cot (cd 25 vuong nho) Trong m o i nho ta ghi mot so' Irich tur tap hdp so
• , , - } Chu'ng minh rang hang, cot va hai du'dng cheo cua hinh vuong thi cd it nha't hai du'dng cd tong cac so' ghi tren ctia nd la bang
Gi§i
Ta cd: So'hang la 5, so'cot la va du'dng cheo la nen chiing gon' ta'tca la 12 du'dng
-T r e n m o i hang hoac m o i cot (hoac du'dng cheo) deu cd chiJa ma m o i cd gia t r i la hay ± nen tong S cac dd phai thoa
- < < nghla la S cd the nhan 11 gia t r i tir -5 de'n
NhuT vay: chi cd 11 gia t r i gan cho 12 du'dng nen cd it nha't du'dng nhan cung mot gia t r i (Nguyen ly Dirichlet)
206
' 8.8: Cac so' nguyen tuT de'n du-dc sap xe'p vao mot hinh vuong
3 x cho tong m o i hang, m o i cot va long cac du'dng cheo deu bang b o i cua Chu'ng minh rang so' ct tam hinh vuong phai la b o i cua
Gi§i
Theo de bai ta cd: 1) A + + C = 9^i, 2) D + E + F = 9/:,,
3) G + // + / = 9/:3,
4) A + D + G = 9A:4,
5) B + E + H ^9k,,
I; 6) C + F + I^9k„ p) A + E + I = 9k,,
8) C + £: + G = 9^,,
Tuf cac phu-dng trinh 2, 7, ta du'dc: E = 9k^_-{D + F)
A B C
D E F
G H I
E = 9k, -{A + I] E^9k,-{C + G)
=^3E = 9k^ + 9/7 + 9k^-{A + D + G + C + F + l)\k ket hdp v d i cac phu'dng trinh 4, 6, la cd:
3E = 9k^+9k, +9k,-9k,-9k, ^E = 3{k,+k, +k,-k,-k,) V a y E la b o i so cua
i {
8.9: Trong hinh vuong x , viet d m o i mot so cho tong bon so' m o i hang bang tong bo'n so' theo m o i cot, bang tong bo'n so theo m o i c u'dng cheo va deu bang a Tinh tong ciia bon so'd bon clinh cvia hinh vuong theo a
(105)Ta gia su* cdc so' diTdc dien vao hinh vuong la:
«11 «12 «13 «M «22 «23 «24
«3I «32 «33 «34
«41 «42 «43 «44
Tong cac s6' d dinh la = a,, + a,^ + + a^,
Theo tinh cha't cac so'ciia hinh vuong nay, ta c6 tong cua 12 so'con lai la 4a - S
Mat khac, siJ dung hai hang dau va cuo'i, ta c6 :
a,2 + + '^42 + ^ 3 = 2a -
Suf dung hai cot dau va cu6'i, ta c6: a^i + "31 + <^24 + ^34 = 2a - Su: dung hai du'dng ch^o ta du'cJc: a^^ + a^^ + a^^ + a^^ —2a —S TO cac phtfdng trinh tien suy ra:
4 a - = a - + a - + 2a-5<i=>4a-5 = a - < ^ = a Vay tong cila bo'n so'd bo'n dinh cila hinh vuong la a
DC' 8.10: Trong moi ciia bang kich thu-dc 25x256, dien vao cac so' +1 hoac - Ta ky hieu tich tat ca cac so'd dong thu" i la a,., tich tat ca cac so'd cot thi? j la bj
ChiJng minh rang : + + + + a^^ + b^^ ^ Goi a la sd'cac so' a bang
b la so' cac so' a bang - c la so' cac so' bj bang d la i; *' "-I'; so' bj bang — 2081
Ta c6 a + b = 25 va c + d = 25
Tich cdc so'trong bang (—l)'' va b^ng (—1)''
Do d6: (-1)" = (-1)^ bwk d cOng tinh ch§:n \6 ^b + d chKn Do dd a^+b^+a^ + b^+ + a^+b^=a-b + c-d
= {a + b)-2b + {c + d)-2d = 25-2b + 25-2d = 2[25 - (Z? + d)\ (vi 25 le va b + d chSn) '
Bi 8.11: Cho da giac loi c6 10 canh
1) Tinh s6'du-dng ch6o
' 2) ChiJng minh rang cd it nhat hai diTdng ch6o tao vdi mot L^ ,.,,,, l<^c_nhohdn^6° ^ '
G i i i
1) Qt ndl hai dinh ciia da giac lai thi c6 mot canh hoac m6t du'dng ch(5o, dd tong s6' canh va drfdng ch6o ciia da gi^c 10 canh 1^:
1 ^ = 45
2
Vay s6'dufdng ch^o Ik: 45 - 10 = 35
2) * Ne'u 35 duTdng chdo c6 hai dtfdng chdo song song vdi nhau thi hai dufdng chdo dd tao gdc 0° (nh6 hdn 6°) * X6t 35 du'dng chdo dd hai dufdng bat chiing la khong song song TO mot diem O bat k^ mat phdng k6 cic dtfdng thing song song Ian lu'dt vdi 35 dufdng chdo cda da gidc thi tao n6n 70 gdc cd tong b^ng 360°
^ 360° , Tacd: = 5°,
70
Do dd cd it nha't mot gdc s6' 70 gdc ndi tren la nhd hon 6° ;
T T T r r m f i
8.12: Cd the c^t da giac idi 17 canh 14 tarn gi^c difdc khong ?
(106)Gidi
Tdng cdc g6c cila mot da giac Idi 17 canh bkng 180° (17-2) = 15.180° , c6n t^ng cac gdc cua 14 tam giac bing 14.180°
Khi chia m6t da giac cac tam giac thi tdng cac gdc ci5a cic tam giac khong the nho hcfn tong cac gdc cua da gidc Vay, khdng the thtfc hien dtfdc phep c^t nhu" de bai
8.13: Trong m6t phdng cd 288 ghe'diTdc xep cac day, moi day diu cd s6' ghe' nhu" Neu ta bdt di day va moi day lai them ghe' thi viTa da cho 288 ngufdi hop (moi ngu-di mot ghe) Hoi trong phdng cd may day ghe' va moi day cd bao nhiau ghe ?
Goi so" day ghe phdng la x (x nguyen du'dng) 288
Moi day cd — ghe, sau bdt di day t u lai x - day (x>2)
288
Luc dd moi day c6n lai cd ghe' x —
Theo bki ta cd phtTdng trinh: _ ^ =
x-2 X
^288x-288(x - 2) = 2x(x - 2) <^ -2;c-288 = GiairadufOc: x^=lS,x^=-l6 (loai)
Vay phdng cd 18 day va moi day cd 16 ghe'
Hi 8.14: Trong mdt buoi hoi thao cd nha khoa hoc tham dU" ba ngtfdi bat k>^ ho thi cd it nhat hai ngu^di ndi cilng mot thu" ti^'ng Hon nffa moi ngffdi cuoc hdi thao ndi dtfdc khdng qua thit tie'ng Chrfng minh cd it nhat ngffdi ndi cilng mot thff tie'ng
•_210
Giii
Ta chffng minh bang phan chffng
Gia su" r^ng khdng cd ngffdi nko ndi cing thff tie'ng
Goi X la mot ngrfdi bat ky cuoc hoi thao Vi x chl ndi dffOc
td'i da thff tie'ng ma thoi va dieu gia tren moi ngtfdi chi ndi cho mot ngtfdi khac cung thff tie'ng nen ngoai x va ngffdi ma X cd the ndi vdi ho cung thff tie'ng thi ngtfdi cdn lai khdng the ndi b^ng thff tie'ng nao vdi ho ca
Goi mot ngu-di ay la y Vi y chi ndi dtfOc td'i da vdi ngffdi nSn ngffdi lai cd it nha't mot ngffdi la z khong ndi chuyen du'Oc vdi y
Luc dd ta cd nhdm (x, y, z) thi: x khong ndi du'Oc vdi y y khdng ndi diTOc vdi z z khdng ndi dffOc vdi x Dieu nky mau thuan vdi gia thie't cu* ngu^di thi cd it nhat ngffdi ndi Cling mot thu" tie'ng
Vay cd it nhat ngu-di ndi cung thff tie'ng
8.15: Trong mot giai bdng da gdm 11 ddi bdng tham dff thi da'u theo the thtfc vdng trdn IffOt di va- IffOt ve (nghia la hai ddi bat ky ludn gap hai tran) va tinh didm bdi: tran thang du'Oc cong diem, tran thua bi truf diem va tran hda la diem
1) Tinh s6' tran thang, tran thua va tran hda cua doi dufOc 13 diem ket thiic giai
2) Ket luan doi dffOc 13 diem la trung binh'cong cua diem cua cac doi la dung hay sai ?
Giii
1) Vdi the thiJc thi dau vdng trdn ItfOt di va vi thi doi phai thi da'u 20 tran
Goi X la sd' tran thing, y la so' tran thua thi ta cd:
(107)7 x - j ' = 13
x + y<20 (2) (1)
Tiir(l)tac6: 6y ^Ix-13 = 6x+ {x-\3) =^ y = x + x-13 x — \3
Dat: / = — - — thi / e Z va tacd: x = 6t + l3,y^7t + l3 (2)chota: 13/ + < o
t<-13 Mat khac X > nan: 6/ + > = ^ r > - 13 V d i - — < / < - — t h i chi CO t = - 6 - Suy ra X = 7, y =
Vay d6i dMdc'U diem se thang tran, thua tran, hoa tran 2) Moi doi ket thuc giai da thi dau 20 tran nen toan giai gdm c6:
20x11 , = 110 tran 2
(Khi ta lay 20x11 = 220thi ta da tinh lap lai hai Ian so' tran dau giffa hai doi n^n phai chia de cd so'tran da'u that suf)
Trong mot tran dS'u ne'u hai doi h6a thi khong c6 di^m nao Trong mot tran da'u c6 thang thua thi tong so' diem ciia hai dpi la bang
Do dd vdi 110 tran thi tong so' diem ciia cac d6i se khong Idn
hdn 110; va nhiT the' xay ^ < 13 tifc la ket luan dpi c6 13 diem
bing trung binh cpng diem cua tat ca cac dpi la sai 8.16: Xet tap hpp A gdm 2005 du-dng thing mat phing thoa cic dac diem sau day:
1) Hai dtfdng thing ba't ky A thi khong song song vdi 2) Ba dufdng thang ba't A thi khong ddng guy Bat lAI la 212
s6' tam giac tao bdi cac du'dng thang A mk khong bi c^t bcti ba't ky mot du'cfng thang khac ciia A ChiJng minh r^ng
A > 2003
ai
• X6l A3 gdm ba dufdng thing thi c6 nhat m6t tam giac can
timtiJc la|A3| = l
• X^t A^gdm bo'n du'dng thing Xdt drfdng thing ghi la thoa tinh chat:
Tat ca cac giao diem cua cac du'dng thing A^ d cung trong mot niJa mat phing ma bd la Gpi I la cac giao diem d gan nha't thi diem I se t^o vdi mot canh d tren a, mot tam giac thoa de bai Ngoai ba du'dng lai da tao mot tam giac can tim niJa nen it nha't cd hai tam giac can tim, vi vay > • tong quit vdi A2005 = A2004 U {^} (d a ngoai A^^ )
Hdn nffa vdi diTdng thing (d) thoa tinh chat cac giao di<^m ciia cac dufdng thing trong A^QOS ci cung mot nufa mat phing cd bd la (d) (luc nao Ajoojta ctlng chpn difdc mot du'dng thing (d) nhu* the ) Gpi J la giao diem d gan (d) nhat thi cd them mot tam giac niTa vdi J la dinh va canh la d tren (d), dd ta cd:
•^005 ^ Tijf cacke'tqua: A3 = 1, A^
2004
>
+
A3 + = 2,|A, > A, +1 = - ^005 > ^004 + suy rang
(108)Vay:
^ 0 51 > (2004 - 2) + = 2003 \A\= Ajoos > 0
8.17: Ta ghi bon so' va nSm s6' l^n trdn mot difdng tron theo mot thu* tu* tily y Sau 66 ciJ giiTa hai s6' bkng ta ghi so' 0, gii?a hai s6' khac ta ghi so' 1; k6' 66 la xoa cac s6' ghi vao luc dau; va lap lai viec lam ban ChiJug minh rkng n6'u ciJ tiep tuc lam mai thi khong the nao nhan du'Oc tren dtfdng tron toan chin so'
Giii
Gia su" la tdi Ian n ta cd du'dc so' tren du-dng tron Suy cl Ian thu" n - cac so' tren du'dng tron phai bkng (va di nhi6n khac 0, vi n^'u da cd so' rdi d Ian n - la mau thuan ) va deu bkng 1, dd d Ian thif n - tr^n du-dng tron phai cd sd'dlu khac ddi mot lien tiep Mud'n dieu xay thi cac so' va cac so' phai bang Dieu la trai vdi d^ bai vi tong cac so'ghi trSn du'dng tron la (day la mot s o l e )
Vay khong the xay tren du'dng tron gdm toan so'O
8.18: Tren du'dng tron ta vi6't 30 so', cho moi so' chung bing gia tri tuyet dd'i cua hieu hai s6'ke'ti6'p theo chieu ngtfdc chieu kim ddng hd Bie't rang tong tat ca cac so' hlng 20 Hay tim ta't ck cac so' ?
Giii
Theo gia thie't ta» suy cac so' da cho la khong am, goi cAc sd'theo iM t\S ngrfdc chieu quay kim ddng hd bat dau tuf so' Idn nha't 1^
â,ậ ,ậ (oj >a,.;i = 2,3, ,30)
Theo gia thie't = « - « mk* < « ^2 ~ ^3 ^'h 214!
odd :
Ng'u ^2 - « <a2=^a^<a^ ma a^ > dd = V d i : = , th^o gia thid't suy
= 0,0^ = a i , a j =ai,ag = Q\ \a^ =a^,a^g = « P« O = , tong cac sd'b^ng 20ai = = > a i =
Do do Oj = = flg = = = ; cAc s6'c6n lai b^ng
• Néu - I < a, ^ â <â ma a, > dj đ =ậ >t V d i : a, = , theo gia thid't suy
=0,a,=a^,a^= 0,a^ = a„ ,a^ = a„a^g = O O J Q = a , Tong c^c s6' b^ng 20a, = 20 =^ a, =
Do dd = ô = ^8 = ããã = ^29 = 0' '^^^ h\n%
8.19: Ba hoc sinh Idp X: Dan, Mao, Thin di chdi, tha'y mdt ngu-di lai xe t6 v i pham luat le giao thdng, khdng nhd s6' xe Ik bao nhi^u, nhufng mSi ngufdi ddu nhd mot dac diem cua s6' xe Dan nhd r^ng hai chff s6' dau gid'ng nhau, Mao nhd Ik hai chff s6' cud'i cdng gio'ng Thin thi qua quye't r\ng so xe cd bd'n chif sd" la mdt s6' chinh phtfdng Chiing ta hay thiJ tim so' xe
Giii
Goi cdc chi! s6' thi? nha't va thif hai \k x, cdc chi? s6' thit ba vk thiJ tiT
J a y Sd'xe se la:
W 1000A: + 100A:+ 10>' + ) ' - 1 0 ; c + !!>' = 11.(lOOx + y) (*) So' chia h6't cho 11 va cung chia h^'t cho 11^ v i theo bki nd Ik
#6t s6' chinh phiTdng NhuT tha' lOOx + y chia ha't cho 11
Dijfa vko dau hieu chia h6't cho 11 thi sd* x + y cung chia h6't cho 11 Bidu dd cd nghia Ik x + y = 11 ,vi moi chff so' x vk y 6iu nhd hdn 10
(109)y = thi X = 5, y = thi x =
Vay ta chi tim s6' xe s6' sau: 7744, 6655, 5566, 2299 Trong b6n s6' cd s6' 7744 = 88Ma so' chinh phuTdng Vay s6'xela:7744
8.20: Trong m6t mang hidi lien lac giQ"a viing A va B c6 n tram
d vilng A va k tram d vi^ng B Moi tram d vung A c6 ihi liin lac
diTdc vdi it nha't k - p tram vilng B ChiJng minh r^ng n6'u: n.p < k thi cd it nha't mot tram d vung B cd the lien lac duTdc vdi moi tram d vilng A
Goi s6' mang hidi lien lac giffa n tram d A vk k tram d vdng B \k s Ta chitng minh bkng phan chiJng
Gia suf khong cd tram nao d viing B lien lac vdi ca n tram d A Ta tha'y 1^:
s<k{n-l) (1)
Mat khdc theo gia thie't thi moi tram d vilng A cd the lien lac diTdc vdi It nha't k - p tram d vi^ng B
Dodd: s>n{k-p) (2) Ta lai tha'y : np < k n6n : nk - np > nk - k, hay:
n{k-p)>k{n-\) (3)
Tir (1) va (2) ta suy : n{k- p)<k.(n-1) trai vdi (3)
Vay phai cd it nha't tram d v^ng B li6n lac difdc vdi ca n tram d vilng A
216
ll/dNG IX: TRICH DE THI TUYEN SINH LOfP 10
DEI
ll/dNG PTTH CHUYEN LE H6NG PHONG 2004 - 2005 J»h^n chpn : Hoc sinh chon mot hai cau sau day :
L la : (4d) Cho phiTdng tfinh: - (m +1) x + 2m' -18 = (c6 an la X)
h) Tim m de phifdng trinh cd hai nghiem deu am
b) Goi ;ci, la hai nghiem ciia phifdng trinh Tim m de cd:
so
X, -X, <
cau lb: (4d) Rut gon cac bieu thiJc sau:
x' +
a) A = p 7=—7
X + yJx+l X-y/x+l
+X+1
2 + yfx yfx -2]iXyfx + X - \fx -I
h) B= p
II Phan bat buSc:
CSu 2: (4d) Giai phifdng trinh:
a) ^3x^ +X-4 =2-2x
(X > 0)
(x>0)
b) 3 - V + 2x) 2x'
^u3: (4d)
- = x +
^ a) Cho A: > 1, y > Chitng minh: x^ly^ + y-Jx^ < xy
b) Cho X > 0, y > va X + y = Tim gia tri nho nha't cua bieu thtfc:
A = 1 - 1
(110)C a u 4: T i m cac so'nguyen x, y thoa he: y
-X — -X - >
y-2 + x + - <
Gi^i
C a u l a :
a) T a c d : A = (m + 9) > ; V m nen phifdng trinh luon cd hai nghiem la: = m — ; :iC2 = 2m +
Phifdng trinh cd hai nghiem deu am A >
m - < ^ 2m + <
m<0 < ^ - ^ m < - m < -
b) T a c d :
<5<;=>m + < < ^ - < m + < < ^ - < m < -
C a u l b : a) Ta cd:
A =
^i^^fx-ij(x + ^fx+lj VI(VJC+I)(A:->/I + I)
x + yfx+l x-sfx+1 -^X-ylx-X-y[x+X + l = i^yfx-lJ
+ X +
b) T a c d : B = ' + ^/7 ^ / I - ;C(N/]^ + I) - (V I + I)^
2 + V]^)(V^ -1 ) - (V^^ - 2)(V^ +1)
^ V r + i ) ' ( V I - i )
• ( V I + i ) ( - i)l
7^
218j
2^ ]fiiZll]_^^(^"^^-2
i :
2 - x >
3; , 2^ ; c - = - x + x '
h) D i e u k i e n :
K h i dd :
•\
9 + 2;c >
^ + 2J C- ^ x^O
^ x = l
2;c' (3 + V9 + 2X \
•\
3 - ^/9 + 2A: 3 - ^ + ^ ) ' (3 + N/9T2^)
- = jc +
2X + : : V T ^ ^ ^ + ( ^ ^ )
9
- ^ + 2x = <^ ^ = - '^''^^''^
^ , i ± ( p l ) = ^ (*)
y , / r ( 7: i) < y - 2
l + _ xy ^^^^
Cong (*) va (*•*) theo ve ta cd: x ^ / y ^ + y^lx-^ <
xy-Dau " = " xay k h i va chi k h i x = y =
b) T a c d : xy < \x + y
1
— —
4 xy
>
(111)_ (^ + 1)(>'+1)A:>> _ (x + \){y + l) _ ^ + ^ + y + i
(xyf xy xy - - ^ + ^ 1 + A > i + 2.4 =
xy xy
1
Dau " = " xay khi x = y = — Vay A =
CSu 4 : Tim cac so' nguyen x, y thoa he
( ) ^
x'-x > (1)
X —X
y-2 + x + l - < (2)
<y-l=^y-\>0^y>l (3)
y-2\<l [ l < ; < ( ) ^ y-2 + x + l <1=^
;c + l < l [ - < A : < Do ta suy xE { - , - , } \h y e {1,2,3
Thijf lai ta difdc tap nghiem can tim la: {(-1;3);(0;2)
(4)
B E
L O P 10 C H U Y E N T O A N
TRl/OfNG T H P T C H U Y E N L E H O N G P H O N G 2004 - 2005
C a u l : (4d)Giaihe: 2x — y x + y 1 2x-y x + y
• = - (/) =
220
(A2: (3d) Cho X > thoa:x^ + \ 1. Tinh :x' + \
L ' * X X
3x
rek 3: (3d) Giai phifcfng trinh: ,
V3;c + 10
= V x + l - l (1)
C3u4: (4d)
a) Tim gia tri nho nha't cua P = 5x''+ 9y^ - 12;c>' + 24x - 4Sy + 82 , , \x + y + z =
b) Timc^cso nguySnx, y, zthoahe , , , „ x'+y'+z =3 Gi§i
CSul : Dieu kien: j c - > ' ^ x + y^O
Dat:
1 u = 2x-y
thi: (/) ^
V = •
3 M - V = - 1 M - v = x + y
2x-y = x + y =
x = y = \'
| : ^ ' + ^ =
$ nen A : ' + - ^ =
1 x +
-X X
- =
1
= 9=^;c + - = 3 {do x > )
X
1 ;c +
-X
X - x - + x'—-x — -\
X X X X
= x' + 1 +
= x' + 1 - - + = 3[49_ 81 = 123
(112)CSu : Dieu kien: 3x + > <^ ;c >
Dat: t = yl3x + l ^ t>0
=3x + l • e + = 3x + 10 Tacd: (1)<^ f ^ = f - l » ( / - l ) f < + l - V f ' +
- ^ =
t = \ (2)
f + l - # T = (3)
(2)<^ V3;c + = ^ ; c + l = l4^;c =
0) ^ yjt^ + t + \ + ^ + 2t + \ 2t =
<^ / = <^ ^ / ^ T T = <S=> 3;c + = 16 <^ X = Vay ( l ) < ^ x = 0V;c =
C a u :
a) Tacd: P = (2x-3>' + ) ' + ( ; c - ) ' + > Dau bkng bat ddng thitc tren xay k h i :
2;c-3>' + 8=:0 y =
x = A
16
x = A
Vay P =
I) Tacd: x' +y' +e ={x + y + z)^-7>(x + y){y + z)(z + x)
^ = 21-?>(x^y){y + z)(z + x)^\^9-{x + y)(y + z)(z + ^(x + y){y + z){z + x)^^^{2,-z){2>-x)(?,-y) = %
Suy - z, - X, - z la cic ifdc so cua Ma cic Mdc so cua la ± , ± , ± , ±
Nhir vay - x , - y , - z nhan mot gi^ tri da neu LSp bang:
222
- +1 -2 -4 -8 - x
^ - y [ - z
Thur tren bang ta diTdc:
x = \ A: = A: = ;c = -
y = ; > = - ; > = z = l z = - z = z =
B E
T R I J N G T H P T C H U Y E N T R A N D A I N G H I A 2004 - 2005
C a u l : (4d) Cho phifdng trinh: / - ( m + 14);c^+(4m + ) ( - m ) = (cd an so" la x)
a) Dinh m de phifdng tiinh cd bo'n nghiem phan biet
b) Dinh m cho tich so' cua bo'n nghiem tren dat gia tri Idn nhat 'Su 2: (4d) Giai phifcfng trinh:
a) x" + 2;c + l| - = 2-x\
\2x-%
b) V'2A- + - V - X =
V x ' +
|^§u 3: (3d) Cho x, y 1^ nai s6' thifc khac Chtfng minh: 2
^ + ^ + > ^y x^ (1) CSu 4: (3d) Tim c^c so' nguyen x, y thoa phtfdng trinh:
Gi§i
C a u l : / - ( m + 14);c\+(4m + ) ( - m ) = (*)
a) E-inh m de phifcfng trinh (*) cd bon nghiem phan biet:
(113)Bat t = x^
(*) <^ - ( m + 14)t + (4m + ) ( - m ) = (* *) / = 4m + 12
r = - m
(*) c(5 bo'n nghiem phan biet : 4m + >
2 - m > - < m < 4m + ^ - m
b) Dinh m cho tich cua bon nghiem tren dat gid tri Idn nhS't T a c d bo'n nghiem cua (*) la ±^,±^[1^, v d i 1^,1^ 1^ nghiemciia(
x^X2X^x^ =t^.t2 = ( m + 12)(2 —m)
= - m ' - m + 24 = - ( m + if +25 < 25Vm =^ Gia tri Idn nha't ciia jCjjCj ^1:3^:4 la 25
difcfc m = — ^ (thda dieu kien d cSu a)
Cau2:
a) x^ + 2x + l - = 2-x^ ^
2-x^>0
x" +|2x + l | - l = - x ' x ' + 2;c + l - l = x ' - 2X + 1| = - J : '
x'<2 2x + \
x'<2 (VN)
3-2x^ >
x'<2
2x + l = - x ' 2X + = X ' -
224
x'<-"
2; c' + x - = 2; c' - x - =
x'<-2 x = x = -\ x =
X = - X =
^2x + 4-2^2-x = 12;c-8
V97 + I
6x-4 X -
yl2x + A + 2s!2-x ^ x ' +
( - < ;c < 2)
_ ^ ~
'42x + + V ^ ] = V9^' + (1)
(2)
(2) =^ (2x + 4) + (2 - jc) +16^8 - 2J:' = 9x' + ^ V - J C ' - X = X ^ -
8 ( ^ - ; c ' - ;c) - 9A:' - 32 ( - - x ' ^ ,
2 V - ; c ' + j r
4x/2
9 x ' - =
2 ^ - x ' + J c = -
JC = ±
-2 ^ 8- x ^ =-%-x (vdnghiemvl : - < x < )
X = ± — M lai ta difcfc = •
(114)C a u :
Dat t = - + ^=>\t\= — X + I ma X + I
y X y X y X y X
> 2 (do bat danc thiJc C6si) ^ I H > ^ ? < - hay < /
K h i d o : = ^ + ^ +
y X
Bat dang thtJc (1) <^ + > 3< <!=^ - 3< + >
4^ ( / - ! ) ( / - ) > (2) (2) la hien nhien ditng t<-2 hay < r' <
x''
Vay : ^ + ^ + > x_^y_ \y
X
C a u : +xy + y^ = ;c'/ =»(2x + 2}')' = (2xy^\)^ -
=> [2xy + + 2x + 2>') [2xy + - 2A; - 2>') =
^ 2xy + + 2x + 2y = 2x}; + - 2x - 2^
•=^x + y^Q
Thay vao phifdng trinh ban dau ta c6:
X = 0, y = hoac x = 1, y = — hoac x = — 1, y =
B E
L(5P 10 C H U Y E N T O A N
T R U C K N G T H P T C H U Y E N T R A N D A I N G H I A 0 - 0 Cau 1: Cho phiTdng trinh: x^ + px + = cd hai nghiem phan biet 01,^2
va phifcfng trinh x^ +qx-\-\ Q nghiem b^,b^
ChiJng minh: (oj -b^){a^ -b^)[a^ + ^2)(«2 + ^2) = 9^ - • Cau 2: Cho cac so' a, b, c, x, y, z thoa: x = by + cz, y = ax + cz,
z = ax + by, x + y + z 2261
Chu-ng minh: — h — ^ + — ^ =
\+a l+b + c
ta) Tim x,y thoa 5x^ +5y^ + 8x>' + x - > ' + = b) Cho cac so' dUdng x, y, z thoa jc^ + + =
ChiJng minh: vrr7 ^/r7 y >
Cau 4: Chvfng minh rkng khong the' c6 cac so' nguy^n x, y th6a phiWng trinh x ^ - / = 9
Gi§i
C a u l :
Theo dinhly V i e t t a c o :
^ + = - ^ ; a^a^=\ b^-\-b.^=-q ; b^b^=\
(«1 + ^ ) ( « +^2)
= (fljOj - ( r t , + rtj+ bl \a^a^ + (fli + 02)^2 + ^2 = (1 + pfo, + ^,^)(l - pb, +bl) = [pb, - qb,){-qb, - pb,)
= {p-(i)K(-p-^)K^^^ -
p^:au2:
-Cong ve vdi ve edc d i n g thiJc ta difdc: x + y + z = (ax + by + cz)
Do x + y-\-z^Qx\Qn ax + by +
cz^Q-Cong hai ve cua cac d^ng thiJc Ian liWt cho ax, by, cz ta diWc:
+ \)x = ax + by + cz \{b-{-\)y = ax + by + cz ;(c + l)z = + fay + cz
X y_ z - + •
1 + a,' \ b ' 1 + c ax + by + cz ' ax + by + cz ' ca + by + cz
x + y + z
ax + by + cz • =
(115)a) 5x''+5y^+8xy + 2x~2y + =
^x + y = 0,x + l = 0^y_^^Q^^^_^
b)
x V T T ^ - ^ ^ _ ^ - 2J:' C h i l n g m i n h tiTdng tiT:
V
C3u :
y =l993^{x-y)[x'+xy + /) = m^
D o 1993 la so' n g u y e n to', n e n ta c6 cac he phiTdng trinh: f j r - ; = 1993
x-y = l
X +xy + y^ = 9 A: + + =
MUC LUC
CHlTOfNG I : D A N G T H l f C C H U O N G I I : B A T D A N G T H l f C
§ : Phep bien doi tifdng diTdng T i n h chat cua ba't dang thiJc § : Bat dang thiJc Cosi (Cauchy)
§ : Bat dang thuTc tarn giac § : Phifdng phap l a m troi
CHl/dNG I I I : SO H Q C
CHUCfNG IV : G I A T R I LCfN N H A T VA G I A T R I NHO N H A T C U A H A M SO
CHUCfNG V : P H U O N G T R I N H
§ : Phifdng trinh bac hai - PhiTdng t r i n h bac ba - D i n h ly V i - e t
§ : Phifdng trinh qui ve bac hai
§ : Phu'dng trinh chufa gia t r i tuyet doi PhiTdng trinh chiJa can thufc
§ : PhiTdng trinh v d i n g h i e m so nguyen
CHl/CfNG V I : H E P H U C I N G T R I N H
§ : H e hai phifdng trinh bac nhat hai an H e ba phuTdng trinh bac nha't ba an
(116)§ : He doi xiJng He dang cap 161 § : He phUdng trinh c6 dang dac biet 169 CHUCfNG V I I : D O T H I CUA H A M SO 179 CHl/ONG V I I I : M O T S O B A I T O A N V E T O HQfP P H E P
D E M D O N G T l f 202 C H U O N G I X : T R I C H D E T H I T U Y E N S I N H LCfP 10 217
230
R E N L U Y E N TOAN NANG CAO DAI SO
* •
TS Nguyin Cam (Chu bien) • ThS Nguyin Van Phifdc
N H A X U A T B A N
DAI H O C QUOC GIA T P HO C H I MINH
Khu 6, Phi/cfng Linh Trung, Quan Thu Dure, TP.HCM DT: 242 181 - 242 160 +(1421, 1422, 1423, 1425, 1426)
Fax: 242 194 - Email: vnuhp@vnuhcm.edu.vn * * *
Chiu trdch nhiem xudt ban
P G S - T S N G U Y E N Q U A N G D I E N
Bi&n tdp
N G U Y E N V I E T H O N G Su'a ban in T R A N V A N T H A N G
Trinh bay bia
M I N H D I E N
TK 01 T(V)
D H Q G H C M - 011/ 304 T.TK.453 - 05(T)
I n 3000 cuon, kho 16 x 24cm Giay ph6p xuat ban 011/ 304/XB-QLXB Cue Xuat bdn cap ng^y 17/ 03/ 2004 Giay trich ngang so: 564 / K H X B ng^y 28/ 09/ 2005