Sáng tạo và giải phương trình, hệ phương trình, bất phương trình nguyễn tài chung

' • Nhu chiing ta da biet phuong trinh, h$ phuong trinh c6 rat nhieu dang va phuong phap giai khac nhau va rat thuong gap trong cac ky thi gioi toan ciing nhu cac ky thi tuyen sinh Dai h

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Quyet dinh xuat ban so: 296/QD-THTPHCM-2013 do NXB Tong Hop

Thanh Pho H 6 Chi Minh cap ngay 19/03/2013

LMnoidau

HQC sinh hoc toan xong roi lam cac bai tap Vay cac bai tap do 6 dau ma ra?

Ai la nguai dau tien nghi ra cac bai tap do? Nghl nhu the nao? Ngay ca nhieu giao vien cung chi biet suti tarn cac bai tap c6 trong sach giao khoa, sach tham khao khac nhau, chua biet sang tac ra cac de bai tap Mpt trong nhimg each do

la tim nhirng hinh thiic khac nhau de dien ta ciing mpt npi dung roi lay mpt hinh thiic nao do phii hop vai trinh dp hpc sinh va yeu cau hp chiing minh tinh diing dan ciia no ' • Nhu chiing ta da biet phuong trinh, h$ phuong trinh c6 rat nhieu dang va phuong phap giai khac nhau va rat thuong gap trong cac ky thi gioi toan ciing nhu cac ky thi tuyen sinh Dai hpc Nguoi giao vien ngoai nam dupe cac dang phuong trinh va each giai chiing de huong dan hpc sinh can phai biet each xay dung nen cac de toan de lam tai li^u cho vi|c giang day Tai lifu nay dua ra mpt so phuong phap sang tac, quy trinh xay dimg nen cac phuong trinh, he phuong trinh Qua cac phuong phap sang tac nay ta ciing rut ra dupe cac phuong phap giai tu nhien cho cac dang phuong trinh, hf phuong trinh tuong

ling Cac quy trinh xay dyng de toan dupe tnnh bay thong qua nhiing vi du,

cac bai toan dupe xay dung len dupe dat ngay sau cac vi du do Da so cac bai toan dupe xay dung deu c6 loi giai hoac huong dan Quan trpng hon niia la mpt so luu y sau loi giai se giiip chiing ta giai thich dupe "vi sao lai nghl ra loi giai nay"

Nhu vay cuon sach nay se trinh bay song song hai van de: Phuong phap sang tac eae de toan va Cac phuong phap giai ciing nhu phan loai cac dang toan ve phuong trinh, hf phuong trinh Diem moi la va khac bi?t ciia cuon sach nay la quy trinh sang tac mpt de toan moi (dupe trinh bay thong qua cac

vi du) va each thiie chiing ta suy nghl, tim ra loi giai mpt bai toan (dupe trinh bay thong qua eae luu y, chii y, nhan xet ngay sau loi giai cac bai toan) Ngoai

ra cuon sach nay con danh ra mpt ehuong (ehuong 5) de trinh bai cac bai toan phuong trinh, he phuong trinh, bat phuong trinh trong cac de thi Dai hpc trong nhiing nam gan day

Tot nhat, doe gia tu minh giai cac bai toan eo trong sach nay Tuy nhien, de thay va lam chii eae ky xao tinh vi khac, cac bai toan deu dupe giai san (tham chi la nhieu each giai) voi nhiing miie dp chi tiet khac nhau Npi dung sach da c6' gang tuan theo y chii dao xuyen suo't: Biet dupe loi giai ciia bai toan chi la yeu cau dau tien - ma hon the - lam the nao de giai dupe no, each ta xir ly no, nhiing suy lu^n nao to ra "c6 ly", cac ket lu^n, nhan xet va luu y tir bai toan

Hy vong cuon sach nay la tai li§u tham khao c6 ich cho cac em hpc sinh kha

gioi, hoc sinh cac lop chuyen toan Trung hpc pho thong, cac em hpc sinh dang

luypn thi Dai hpc, giao vien toan, sinh vien toan cua cac tmong DHSP,

DHKHTN cung nhu la tai phyc v\ cho cac ky thi tuyen sinh D^i hpc, thi hpc

sinh gioi toan THPT, thi Olympic 30/04

Cac ban hpc sinh, sinh vien, giao vien va nhirng nguoi quan tarn khac se c6

the a m tha'y thieu sot a cuon sach nay trong qua trinh su dung Do vay, su gop

y va chi trich tren tinh than khoa hpc va huang thien tu phia cac ban la dieu

chiing toi luon mong dpi Hy vpng rang tren buoc duang tim toi , sang tao

toan hpc, ban dpc se tim dupe nhiing y tuong tot hon, mai hon, nham bo sung

cho cac y tuong sang tao va loi giai dupe trinh bay trong quyen sach nay

Tac gia

Thac sy: NGUYEN T A I CHUNG

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ChiMng 1 PhUcfng phdp sang tdc va giai phUcfng trinh, hf phUOng trinh, bd'i

1.2 Phifdng phdp duTa ve h$ 5 1.3 Phufdng phap diTa phiTdng trinh ve phifdng trinh ham 15

1.4 Mot so phep dSt an phu cd ban khi giai h$ phufdng trinh 26 1.5 PhiTdng phdp cpng, phufdng phdp the 35 1.6 PhiTdng phdp dao an Phifdng phap hiing so bien thien 54

1.7 PhiTdng phdp sijf dung dinh li Lagrange 63 1.8 Phu'dng phdp hinh hpc 69 1.9 PhiTdng phap ba't dang thtfc 82 1.10 PhiTdng phap tham bien 95

ChUcfng 2 PhUcfng phdp da thiic va phUcfng trinh phdn thitc hOu ti 116

2.1 Cdc dong nha't thiJc bo sung 116 2.2 PhiTdng trinh bac ba 117 2.3 Phu'dng trinh bac bon 127 2.4 PhiTdng phdp sdng tdc cdc phiTdng trinh da thiJc bac cac 137

2.5 PhiTdng trinh phan thiJc hi?u ti 149

ChUcfng 3 PhUcfng trinh, bdtphUcfng trinh chiia can thiic 158 3.1 Phep the trong doi vdi phiTdng trinh 3/A(X) ± }JB{X) = 3^C(x) 158

3.2 PhiTdng trinh (ax + b)" = pJ^a'x + b' + qx + r 160

3.3 PhiTdng trinh [ f (x)]" + b(x) = a(x)!i/a(x).f (x) - b(x) 168

3.4 PhiTdng trinh ding cap d6'i vdi ^P(x) v^ ^Q(x) 174 3.5 Phu'dng trinh doi xiJng d6'i vdi ^P(x) vd ^ Q ( x ) 179 3.6 Mpt so hiTdng sdng tac phiTdng trinh v6 ti 184

ChiMng 4 //# phUcmg trinh, h? bat phUOng trinh 231

4.1 He phiTdng trinh doi xuTng 231

4.2 He c6 yeu to d^ng cap 253

4.3 H$ bac hai tdng qu^t 266

4.6 SuT dung can bac n cua so phuTc de sang tac va giai he phiTdng trinh

307 4.7 Phi/dng phap bien doi ding thiJc 314

4.8 MotsohekhongmaumiTc 317

ChUctng 5 Cdc bai todn phUcmg trinh, h^ phUcfng trinh, bat phUcfng trinh trong

dethidt^ihQc 328

5.1 Phtfdng trinh, bat phi/dng trinh chiJa can 328

5.2 He phiTdng trinh dai so 332

5.3 PhiTdng trinh liTdng gidc 337

5.4 Phi/dng trinh, bat phi/dng trmh c6 chlJa cdc so n!,Pn, A^, C\5

5.5 PhiTdng trinh, ba't phiTdng trmh mu 368

5.6 PhuTdng trinh, ba't phifdng trinh logarit 373

trinh, bat phi:^dng trinh

Trong chitcJng nay ta se trinh bay nipt so phUdiig phap cO ban va mot so phUdng phap dac biet di giai va sang tac phitdug tiinh, lie phUdng trinh,

bat phUdng trinh Co mot so vi du, bai toan c6 sii dung den kien thi'tc cua

plntdug trinh da thi'tc bac ba, ban doc c6 the xcni bai i)liitdiig tiiiih bac ba d chUdng 2 (chi can c6 kign thv'tc ve lUdng giac la co the hieu bai phUdng trinh bac ba) trudc khi xem cac bai toan, vi du nay - , •

1.1 PhifcTng phap he so bat dinh

PhUdng ])hap he so bat djnh la chia khoa giup ta i)han tich, tim dudc l a i giai cho nhieu locii phUdiig trinh Chung ta se Ian lUdt tini hieu phUdng phap nay thong qua cac bai toan va cac km y ngay sau do

B a i toan 1. Giai phiCdng trinh 2^^ - l l x + 21 - 3^4.x- - 4 = 0

G i a i Tap xac dinh D = E Plntdug trinh da cho tu'diig ditdng vdi

K h i do dirdc phitdng trinh doi vdi u, v c6 thg phan tich ditdc

V i d u 1 Ta sc sang tdc mM phUdng trinh duclc gidi hhng phiMng phdp he

so bat dinh nhu sau : Ta c6

{a-b+ l ) ( 2 a - 6 + 3) = 0 ^ 20^ + 6^ - 3a6 + 5a - 46 + 3 = 0

Tii day lay a — ^Jl + x vd b = \/l - x ta diidc

2x + 2 + 1 - X - 3 \ / l - x2 + 5VI+X - 4s/l-x + 3 = 0

Rut gon ta duac bai toan sau

B a i toan 3 Gidi phuang trinh 4 \ / l - x = x + 6 - 3 \ / l - x^ + 5s/TTx

D a p so X = - - — - J

B a i toan 4 Gidi phuang trinh 4 + 2\/l - x = - 3 x + SVxTT + V l - x^

D a p so PhUdng trinh c6 tap nghiem S — I 0; — ; — I tah s v , ;: i ;

• Vdi u - 2t; = - 3 , ta c6

Vay phiMng trinh c6 tap nghiem 5 = < 1; ——^ I ,

Lvfu y Phudng phap he so bat dinh de giai he phudng trinh se ditdc de cap

trong phan phan tich t i m Idi giai cac bai toan ciia bai 1.5 : PhUdng phap cong, phiTdng phap the (d trang 35)

1.2 Phifcfng phap difa ve he

Dg giai phUdng trinh bang each dua ve he phUdng trinh ta thutdng dat an

phu, phep dat an p h u nay cimg vdi phUdng trinh trong gia thiet cho ta mpt h$

phitdng trinh Sau day ta se trinh bay phuldng phap sang tac (thong q u a cac

V I du), phUdng phap giai (thong q u a Idi giai cac bai toan va quan trong hdn

niia la cac hru y sau Idi giai) Cac phudng phap sang tac ciing nhiT phifdng

phap giai cac phUdng trinh bang each dUa ve he con dildc de cap rat n h i i u

6 sau bai nay (chang h a n bai 3.2 d trang 160)

V i d u 1 Xet I y ~ 2 ^ 3^^ ^ x = 2 - 3 (2 - 3x•'^)^ Ta c6 hdi todn sau

2 1-V21 1 + V21

X = - 1 , X = - , X = — — , X =

3 6 ()

Lxiu y T i r Idi giai t r c n t a thay iftng neu kliai Irien (2 - 3x'^)'^ tlii sc dira

phitdng t r i n h da cho vc phiWng t r i n h da thiitc bac bon, sau do Ijien doi t h a n h

(x + l) ( 3 x - 2)(9x2 - 3x - 5) = 0

Vay ncu k h i sang tac de toan, t a c6 y lam cho p l n M n g t r i n h khong v6 nghiem

h i i u t i t h i p h i l d n g phap khai t r i e n dua ve phUdng t r i n l i bac cao, sau do phan

t i c h dua ve phu:dng t r i n h t i c h se gap nhieu klio khan

V i d u 2 Xet mot phucing trinh bac hai c6 cd hai nghiem Id so v6 ti

5x'^ - 2x - 1 = 0 ^ 2x = 5x2 _

2x = 5 5 x 2 - 1 \ - 1 Ta CO bdi todn sau

B a i t o a n 7 Gidi phUdng trinh 8x - 5 (5x2 _ _ _ ^

Vdi y = X , t h a y vao (1) t a dUdc 5x2 _2x -\=i) ^ x =

L u t i y Phep dat 2y = 5 x 2 - 1 chrdc tini ra nhu sau: Ta dat n:y-\-b = 5x2 _ ^

vdi a, h t h n sau K h i do t i i u dUdc he

G i a i Dat 7y = 5x^ + 17, ta c6 h? phitdng trinh

J7y = 5x2 _ 17

1^5,y2 - 343a; - 833 = 0

7y = 5x2 _ 17 (1) 7x = 5 y 2 _ 1 7 (2)

Lay (1) trit (2) ta c6 7{y - x) = 5(x + y)(x - y) ^

* Neu X = •(/, thay vao (1) : Sx^ - 7x - 17 = 0 x =

* N i u 5x + 5j/ = - 7 , ket hdp (1) ta c6

x = y 5x + 5y = - 7

\x = 8?y3 - v/3 (2) Lay (1) t i l t (2) theo ve ta dudc

6(y - x) = 8(x3 - 2/3) ^ (a; _ [g (x^ + xy + y^) + 6] = 0

V i x^ + xy + y2 > 0 nen 8 (x^ + xy + y^) + 6 > 0 Do do t i t (3) t a dUdc x =

Thay vao (1) ta dUdc

ay + 6 = 8x3 - v/3 162x + 27V3 = a3y3 + 3a26y2 + Safety + ^3

Can chgn a va 6 sao cho : 8 73

162 o? 27\/5 - 63 3a26-3a62 = 0

Vay ta c6 phep dat 6y = 8x3 _ ^

V i d u 4 Xet tarn thiic bdc hai luon nhdn gid tri duang : x'^ + 2 Khi do

/ x^ + 2) dx = — + 2x + C

^ 3

x3

Chi cdn chon C = 0 ta diicfc mot da thiic bdc ba dSng bien la h{x) = — + 2x

Ta CO /i(3) = 15 Vdy ta thu duoc mot ham so da thiic bdc ba dong bien g{x)

A;(x) = y tuang dudny U(H • — + 3x - 15 = y <^ x^ + 9x - 45 = 3y TH phiiang

tnnh cuoi ndy thay x bdi y ta thu duac he doi xilng loai hai

x3 + 9x - 45 = 3y y3 + 9y - 45 = 3x

Til: he tren, sU dung phep the ta thu duoc phuong trinh

x3 + 9x - 45

+ 9 / x 3 + 9 x - 4 5 \ - 45 = 3x 0fiOJ i.'

9

d a t x^ + 6 x ~ 45 - 3y, t a so t h u ducJc m o t ho d o i x i ' m g loai h a i

V i d u 5 Chon mot phMng trmh chi c6 hai nghiem / d 0 v d 1 IdlV = l O x + 1

Tii phiCOng trinh nay ta thiel lap mot he doi xvCng loai hai, sau do lai quay

ve phuang trmh nhu sau :

f l l ' - = lOy + 1 ^ I y = l o g n ( l O x + 1) - = l o g , , ( l O r + 1)

\ i r y = 10x + l ^ \ l F = 1 0 y + l ^ 10 l o g i i u u x + i j

Suy ra I F = l O l o g n (10x + 1) + 1 ^ I F = 2 1 o g i i ( 1 0 x + 1)^ + 1 Ta c6 bdi

todn sau

B a i t o a n 1 1 Gidi phuang trinh IV = 2 1 o g i i ( l O x + 1)^ + 1

G i a i D i c u kiCm > D a t y = logn ( l O x + 1), Ivhi d o I P = 1 0 x + 1 K e t

hdp v d i phUdng t r i n h da cho, t a c6 he | JJy ^ ^ | ^ "» " • - - w.,

V i d u 6 Ta se su dung phuang phdp lap di sang tdc phuang trinh tit he

phuang trinh doi xvtng loai hai Xuat phdt tit \^^Z ^^^4^30 •^^ ^^'^'"'(1 P^^'^P

the ta diMc phuang trmh Ax = ^ 3 0 + | v / x + 30 Til phuong trinh nay ta lai, thu dtWc he doi xtCng loai liai

Ta CO bdi todn sau

B a i t o a n 12 ( D e n g h i O l y m p i c 3 0 / 0 4 / 2 0 1 0 ) Gidi phuang trinh

4x = 30 + - W 30 + - ^ 3 0 + - \/^T30

G i a i Do x la nghifni t h i x > 0 Dat u = 30 + ^x + 30, tit phitdng t n n h

4v = Vx + 30 > VtTTSO = 4a: 4u > 4a: =^ V > a; =^ u = X

, f T > 0 1 + \/l921

Vay r = x va 4.T = ^ | Jgp^^ ^, ^ 30 — PhUdng

trinh da cho co nghiem dny nhat x = 1 + 71921

32

V i d u 7 Vdi X = 8 thi ^/x-\-8-\- \ J x - l — 3, ia c6 bai todn {ch&c chan co

mot nghiem dep x = 8) sau

B a i t o a n 13 Giai phUdng trinh y/x + 8 + \/x - 7 = 3

^ ro < 1/- < 3 ^ ro < u < 3

^ \4u^ - 18u2 + 36u - 32 = 0 ^ \ = 2

-Tit do ta t h u dudc 1 = 2 <=>{^ + f=p ^ x = 8 (thoa man

dieu kien) Vay phitdng trinh da cho co nghiem duy nhat x = 8

< 3

u2 + (3 - uf = 5 ^ '

L t f t i y- Doi vdi phu'dng trinh - JJx) + "\/b + / ( x ) = c, t a co each giai :

Dat u = 'ija - fix), v = '^h + f(x), dan den he { ^ H ^+,n"s'^['^ Nhit vay

dang nay la j)hn'dng trinh vo t i , infi san khi dat an phu dita ve he, roi dimg phep the dan tdi phudng trinh da thitc, do do k h i sang tac de toan ta phai

dac biet chii y cac chi so can Chang han d v i dn 7 t h i m = n = 4 nen t a yen

tam rang se dan tdi phitdng trinh da thite bac 4 co i t nhat mot nghiem dep

V i d u 8 Vd'i x = - 2 thi 2<y3x -2 + 3^6 - 5x = 8, ta c6 bai todn {chac

chdn CO mot nghiem dep x = - 2 ) sau 1 "i;.,/ ;- f '-,1,;

B a i t o a n 14 Gtdi phiMng trinh 2 v^3x - 2 + 3^6 - 5x = 8

G i a i Dieu kien x <^ Dat u = ^ 3 x - 2, v = ^/G - 5x > 0 K h i do 5

B a i t o a n 15 Giai phiMng trinh

1 + \ / l - x2 [ V ( l + :r)-* - ^ ( 1 - x)'A^ =2+ yjl - xK ffif, t uM

G i a i Dieu kien - 1 < x < 1 Dat ^l + x = a, \ / r ^ = vdi a > 0, 6 > 0

K h i do a' + l? = 2 Ta co he sau ( \ S "" '

\l + ab{a-^ -b^) = 2 + ab (2) ,

(1) =^ {a + bf = 2+ 2ab^ s/lT^=-^{a + b) [do a,b>0)

V2 Ket-hop (2) ta co ' , |

1 1 ' ( '

-7= (a + b){a - b){a^ + b^ + ab) = 2 + ah => ^ ( a ^ ~ h'-) = I

v 2 v 2 Tit do t a c6 he | ~ ^2 ! l 2 ^ Cong hai phitdng trinh ve theo ve t a co

2 a 2 - - = 2 + y 2 ^ a 2 = l + 4 = ^ l + a ; = l + ^ ^ x =

Vay phitdng trinh co nghiem duy nhat x = — vj / j

B a i t o a n 16 Gidi phuang irinh \/\/2 1 - x + =

8 +

3 ' 18 nen nghiem duy nhat ciia phudng trinh la

ta CO the sang tac va giai dUdc nhien phitdng trinh hay va kho, thudng gap

trong cac k}' t h i hoc sinh gioi D6 van dung dildc phitdng phap nay, ta thirdng

bien ddi phiWng t i i n h da cho thanh phitdng trinh ham f {<f){x)) = f (ipix)),

trong do / la ham ddn dicu Tfr day diui den mot phvtdng trinh ddn gian lidn

4>{x) = tpix). De giai dUdc cac bai toan bang phitdng phap nay t h i nhftng kien thifc ve ham so nlut dao ham, xet sit bien thien va k l nang doan nghiem

la cite k i ciuan trong, c6 nhitng bai doan dUdc dap so la da hoan thanh den hdn 90% Idi giai Phitdng phap nay ditdc si't dung nhien, chang han d muc

3.6.3 d trang 191 M o t so tritdng hdp dac biet thitdng gap :

• Neu / la ham ddn dieu tren khoang (a; 6) t h i plutdng trinh / ( x ) = k {k la hang so) CO khong qua 1 nghiem tren khoang {n; h)

• Neu f yk g \h hai ham ddn dieu ngitdc chieu tren khoang (a; b) t h i phitdng trinh / ( x ) = g{x) c6 khong qua 1 nghiem tren khoang (o;6)

• Neu ta thay cum t i t " / la ham ddn dieu tren khoang (a; 6)" bdi cum tit

" / la ham ddn dieu tren m5i khoang (a; 6), {c]d)" t h i hai ket qua d tren se

khong dung, ti'tc la plutdng trinh co thc^ sc c6 nhien hdn mot nghiem Ban

doc hay xein bai toan 20 d trang 16

B a i t o a n 18 ( H S G Q u a n g N i n h 2 0 1 1 ) Gidi phieang tnnh

1 1

= +

v^5x - 7 v / ^ ^ = 0 (1)

B a i t o a n 19 ( H S G L a m D o n g , n a m h o c 2010-2011) Gidi phuang trinh

G i a i Dieu kien x > 1 Dg thay x = 1 khong la nghigm ciia phirong trinh nen

Do do ham so n a y d o n g bien Suy ra (*) c6 khong qua mot nghiem, mat khac

f{2) = 7 nen phitdng t r i n h d a cho c6 nghiem duy nhat la x = 2

' B a i t o a n 20 Gidi cdc phiCcfng trinh

Phitdng t r i n h da cho c6 hai nghiem x = - 2 , x = 1

L i r t i y Phep phan tich (2) difdc t i m ra n h u sau : Ta can t i m sao cho

- (x^ - 3x + 2)=k [(x^ + 2x) - (2x^ - X + 2)] =^k = l

Con phitdng t r i n h t6ng quat a-^(^) - a^^^) = h{x) dUdc giai tUdng t u Thudng

t h i t a dung plntdng phap he so bat (Hnh u h u tren de difa vc

,^ hix) = k[g{x) ^ fix)] ,

B a i t o a n 22 ( H S G T h a i B i n h n a m h o c 2010-2011) Gidi phUdng trinh

2 x - 1 (x - 1)^

V i /'(/:) = — ^ + 1 > 0 , > 0 neii / (long bicii t i c i i (0; +oo), tit (3) c6

/.In 3

\2 ^ o 2

2x - 1 = 3(a; - 1)' <^ 3x' - 8x + 4 <^ x e | 2 , ? | (thoa di^u kien)

Tap nghiem ciia phudng trinh (1) la 5 = | 2 ,

Lifti y Pliep phan t i d i (2) dUcJc tini ra nhit sau : Ta can tini n, ft, 7 sao cho

Dang tong quat log„ —T-T- = h(x) dUdc giai titdng tu Ta thitcing dung phudng

phap he so bat dinh nhir tren dfi dua ve mot trong cac tnrdng hdp

h{x) = -fix) + a'=g(x) + k, hix) = fix) - a'gix) + k, /i(x) = k[gix) - fix)]

Bai toan 23. Giai phucing trinh Sx^ - SGx^ + 53x - 25 = \/3x - 5 (1)

y-• Bai toan nay con c6 each giai khac, dirdc de cap 6 bai toan 8 d trang 163

• Do ve trai c6 bac 3 con v6 phai co bac - nen ta can difa 2 ve ve bieu thiic

o

dang fit) = mt'^ +nt De y rang hang tit v'3x - 5 6 ve phai c6 bac thg,p nhat

nen no tiidng ling vdi nt trong / ( i ) , vay n= 1 Lifu y r^ng 8x^ = 8(x^) = (2x)^

nen d day ta phai xet 2 trUdng hdp, m = 8 hoSc m = 1 Ngu m = 1 thi

fit) = t^ + t Do do can dita (1) ve dang

(2x - M ) 3 + (2x - u) = 3x - 5 + ^3x - 5

<^8x^ + x2(-12u) + x(6i/2 -l)-u^ -u + 5= \/3x - 5

Dong nhat he so vdi ve trai cua (1) ta dildc

-12u = -36 : *' 6w2 _ 1 = 53 u = 3 s

-u^-u + 5 = -25

Vay trifcJng hdp rn = 1 da cho ket qua, do do khong can xet m = 8

• Nhiing budc phan ti'ch tren nhhi tuy dai nhung khi da quen roi tlii ta c6

t h i tinh rat nhanh Tuy nhien, trong mot so bai toan, ham f { t ) khong dong

bien tren R nhitng ta co th^ chi can xct ddn dicu tren mien xac dinh D

Bai toan 24. Giai phuang trinh 9x2 _ 28^; + 21 = sjx - 1

(1) <^ (4 - 3x)2 + 4-3a; = x - l + \/x - 1

^ fii - 3x) = fi^/^i^) (vdi fit) =t^ + t)

4 - 3x = i / x - 1 ^do f { t ) dong bien tren ( - ^ ; +c)o)^ •

r 4 - 3x > 0 " - 3 25 - \ / l 3

1 ( 4 - 3 x f = x - l

4 , 2 5 ± \ / l 3 «>x = 18

18

Vay (1) CO tap nghiem S'= {2; ^ ^ - j ^ ^ }

Lxiu y Ta xay dvtng ham fit) = mt^ + nt Dg y rftng ha,ng t i i v/x - 1 d v6

phai CO bac thap nhat nen n = 1 V i 9x2 _ 9 (^.2^ ^ l.(3x)2 nen ta phai xet

2 tritdng hdp m = 9, m = 1

• Ngu m = 9 t h i / ( / ) = 9/2 + / Can dua (1) ve dang

9(x - 'u)2 + X - u = 9(x - 1) + v/x - 1 -»9x2 + x ( - 1 8 u - 8 ) + t i 2 - u + 9 = v ^ x - 1

Dong nhat he so t a dUdc:

/ - 1 8 w - 8 = - 2 8 ^ j i i = — f s

• Neu 771 = 1 t h i fit) = t'^ + t Ta can diia (1) ve dang

(3x - uY + 3x — u = {x — 1) + \/ X — 1

<^9x^ + x{-6u + 2) + - u + 1 = y/x - 1

Dong nhat he so t a duoc | ^^^1'^^'^'^" = 5 Den day c6 le bai

toan d a ditdc giai quyet, nhung that r a "chong gai" con ci phia trudc

khi 3 x - 5 > - - < ( ^ x > - Con 1 < x < - t h i sao ? Lai de y rang ham so

bac 2 cung c6 cai hay cua no, do la ( - / ) 2 — tren, dua vao he so bac cao

nhat la 9, t a chi mdi xet t = 3x - u nen bay gid t a se xet i = u - 3x Can

toan mdi thuc sU dudc giai quyet N h u vay t a can linh,hoat trong viec xay

dung ham so, nhat l a doi vdi ham bac chan Ta cuiig c6 the giai bai toan

tren bang each dat ^x-\ 3?/ - 5 de dUa ve he doi xi'tag loai 2

B a i t o a n 25 Giai phuong trinh

Day l a phUdng trinh da thUc bac 3, dUdc do cap d bai 2.2.1 (d trang 117)

L i i u y Nhan 9 cho 2 ve ciia (1) t a dUdc

27X''' - 54x2 2 7 3 153 = 2 7 ^ 9 ( - 3 x 2 + 21 + 5 ) ' (2)

Do bieu thiJc chita can c6 he so la 27, hang t i i bac cao nhat la 27x^ = (3x)^

nen ta se difa hai ve ciia (2) ve dang f{t) = t^ + 27t Ta ])ien doi (2) thanh :

Co the ban doc se thac mac tai sao lai nhan 9 ma khdng phai la so khac

That ra dieu nay da dUdc de cap den roi K h i xay dUng ham f{t) = mt^ + 3t,

ta thudng nghi tdi 3x'^ = 3(x^) nen cho m = 3 ma quen r i n g ngoai ra con c6

3x'' = Q(3.X)^ (trudng hdp nay that ra hiem gap) N h u vay f{t) ciing c6 the

la - + 3 i Viec nhan 9 chi ddn gian la k h i i mau so. S i ' s s i ; : ,,, f ; , ?

B a i t o a n 26 ( D e nghi O l y m p i c 3 0 / 0 4 / 2 0 1 1 ) Giai he phxcang trinh

30.7:2011 ^ 4^^,^2010 ^ 30y4022 ^ 4y2012 , , „

1627/2 + 27 ^ 3 = ( 8 x ^ - ^ 3 ) ^ (2)

G i a i Thay 7/ = 0 vao he thay khong thoii man, vay chi xet y 7^ 0 Ta c6

™2011 „ ( 1 ) ^ 3 0 ^ + 4 - = 30^2011 + 47/ (3)

y y

Xet ham so f{t) = 30<20ii ^ 4^^^^ ^ y/^^^ ^ 3 0 2 0 i u 2 0 i o + 4 > Q nen

ham / dong bien tren M Do do t i f (3) ta c6

Then l)ni loan 9, cJ trang 8, cac nghiem cua (4) la cos;^ cos^i^ c o s i ^

-lo 18 18

Do cos —— < 0, ('OS - — < 0 nen ta chi nhan nghiem x = cos — Cac nghiem

18 18 18

cua he phuring trinh da cho la ^ ,

Bai toan 27 (De nghi Olympic 3 0 / 0 4 / 2 0 1 1 ) Gim phuang trinh

Hu'dng dan Xet, /(,;:) = Vx'-^ + 3.7-2 + g^ _ 13 _ ^^332^ Ta c6

nghiem duy nhat cua phitdng trinh da cho

Bai toan 28 (De thi hoc sinh gioi cac trtrdng C h u y e n khu vu'c D u y e n

Hai va Dong B a n g B a c B o nam 2010). Gidi phiMng trinh

2.x'* - x2 + v/2x^ - 3 x + 1 = 3 x + 1 + v/x2 + 2

Hu'dng dSn Tap xac dinh D = M Bien d5i phildng trinh ve •

2x'* - 3x + \/2x'-^ - 3 x + 1 = x2 + 1 + \/x2 + 2

Xet ham ,s6 / ( / ) = t + <yt + I tren R De chitng minh ham so dong bien tren

R Tvt gia thiet suy ra

Xet ham so dac trung f{t) = t'^ + t, t eR Ta co / ' ( i ) = Si^ + 1 > 0, vay ham

so dong bien tren R nen (*) <=i> a + 1 = Ta co he sau :

Sii dung phep the ta co ; ; ;

^3 _ _ 1)2 = l<=^ 6'^ - 362 + 66 - 4 = 0 o (6 - ^^(,^2 _ 26 + 4) = 0 ^ 6 = 1

Tit do X = - 1 Vay phUdng trinh co nghiem duy nhat x = - 1

I'

1.3.2 Phu'dng phap sang tac bai toan mdi -' ^

V i du 1. Xudt phdt tic mot phuong trinh cd each gidi rat cd ban, do la :

- r x - i = 6x - 5 Xet mot ham so dOn dicu (j){t) = t + 6 logy t Khi do

Ta duoc bai toan sau

Bai toan 30. Gidi phuang trinh 7'^''^ = 1 + 2 logy (6x - 5)^ (1)

C a c h 1 De thay r i n g x = 1, x = 2 thoa ( 4 ) Xet ham f{x) = T'^ - 6x + 5

tren R Ta co / ' ( x ) = 7 ^ - i l n 7 - 6; / " ( x ) = 7^-^ (In7)^ > 0, Vx G R Vay ham / CO do thi luon luon loin nen cat true Ox tai khong qua hai diem, suy

ra (4) CO khong qua 2 nghiem Vay x = 1, a; = 2 la tat ca cac nghiem cua

(4) Phitdng trinh (1) co tap nghiem la 5 = {1,2}

C a c h 2 Ta co / ' ( x ) = 0 7^-' = — x = xo = 1 + logy (6 logy e) V i v6i moi x G R thi / " ( x ) > 0 nen suy ra / ' la ham dong bien tren R va

/ ' ( x ) < 0, Vx G ( - 0 0 ; xo) ; / ' ( x ) > 0, Vx € (XQ; + 0 0 ) ,

Vay ham / nghich bien tren {-OO;XQ) va dong bien tren (xo;+oo), do do

(4) CO khong qua 2 nghiem Vay x = 1, a; = 2 la lat ca cac nghiem ciia (4)

PhUdng trinh (1) c6 tap nghiem la S = { 1 , 2 }

L i f u y Phep phan tich (2) diidc t i m ra nhil sau : Can chon a, /3, 7 sao cho

l = a ( x - l ) + / 3 ( 6 x - 5 ) = { - + 6 / ? j 0 ^ ^ { ? = T '

Cac phiTdng trinh long quat

' ' ' " ' - klog,jj{x) = h{:r.) (vdi a > 1, A; > 0

a-''^^) + klog^gix) = hix) (v6i 0 < a < 1, A: > o)

Dudc giai tudng tif nhil tren bang phitdng phap he so bat dinh, phan tich

Ta CO bai toan sau

B a i t o a n 31 Gidi phudng trinh 2x (v/3) ^ - 2x ( -1\ = 2x2 _ 2x - 1

Hu'dng d a n Tifdng tif bai toan 21 d trang 17 PhiTdng trinh c6 hai nghiem

l + V^ l - \ / 3

X = — - — , X = — - —

V i d u 3 Xet ham so nghich bien tren khodng (0; + 0 0 ) la f{t) = logi t — t

Tii phuang trinh ham j (\{x - = f {2x + \) ta cd

Ta diCcfc bdi toan sau ^ 1 ' •••

B a i t o a n 32 Gidi phuang trinh 8 logi -~—-j- = x'^ ~ ISx - 31

Hii'ding d a n Tudng t i t nhxi bai toan 22 d trang 17 Phitdng trinh c6 hai nghiem x = 9 - 2^22,x = 9 + 2\/22 /^y i^ ;^ .y) Q V,

B a i toan 33 ( D e nghi O l y m p i c 3 0 / 0 4 / 2 0 1 1 ) Gidi phuang trmh

9x2 + (ix + 3126 x^ - 2x = 1 + ^ / ^ + in ^ , ^ 3 ^ ^ ^ , ^ • •

Htfofng d a n PhiJdng trinh viet lai '• ' '' "' <f 1 - J ''

/ ( x ) = / ( A / S ^ T T ) , vdi / ( i ) = + i + ln(^« + 3125)

T a c 6 / ' ( i ) = 3i2 + i + - ^ J ^ i _ , ma g«l.>lifii|

5(t*^ + 3125) = Si** + 5*^ > 6^56^30 = 30

+ 3125 < 1 nen ham c6 / ' ( < ) > 0, suy ra ham / dong bien tren E -i •

B a i t o a n 34 ( D e nghi cho ki t h i hoc sinh gioi cac tru'dng C h u y e n

khu v y c D u y e n H a i v a D o n g B a n g B a c B o n a m 2010) Gidi phudng

trinh

(6^' - 3^) (19-^- - 5^) (10^ - r) + (15^ - 8^) (9^ - 4^') (5^ - 2^) = 2 3 F (1)

G i a i Ta c6 cac nhan xet sau :

N h a n xet 1. Vdi a > 6 > c > 1 t h i > 6^ neu x > 0 va < 6^ neu x < 0

N h a n xet 2. Vdi a > 6 > 0 cho tritdc t h i ham so / ( x ) = - 6^ xac djnh

dong bien va lien tuc tren tap D = fO; + 0 0 ) do

trong k h i 231'^ > 0, auy ra phitdng trinh khong c6 nghi^m khong ditdng

V6i X > 0, chia hai vg phitdng trinh cho 2 3 F = (3.7.11)^ dudc

Goi y la hani so d ve trai ciia (2) T i t nhan xet 2 va nhan xet 3, suy ra y

dong bien tren D = (0; +oo) va

11 7 ^ 1 1 ' 7 3 ~ •

Vay (1) ^ g{x) = y{\) o x = 1 Suy ra x = 1 la nghiem duy nhat ciia (1).'"'I

1.4 M o t so phep dat a n phu cd ban k h i giai he

phufdng trinh

Co rat nhieu each dat an phu k h i giai he phitdng trinh Dat an phii nhit the

nao con t u y thuoc vao tifng he phitdng trinh cu the Bai nay se neu ra mot

so phcp dat an phu cd ban, thit6ng gap N a m ditdc cac phep dat nay t a se

CO dinh hudng tot hdn khi giai he phitdng trinh

Thay vao he t a ditdc he doi xiing loai 1 doi vdi u va v { ""2"^^ t ^

B a i t o a n 36 Gidi he phitang trinh | ^ J ^2 _ ^ _ ^^^^

Hu'o'ng d i n D a t u =^ x + y, v = x - y- K h i do

u +-V = 2x, uv = x2 - y2, ^2 + 1-2 = 2(x2 + y^)

Thay vao he ta ditdc | ^f+^;^ I T i t (1) suy ra u

thav vao (2) dUdc :

uv (^2 + t'2) ^ u +

01;

[uvY = - 5

Neu u = 0 t h i y = - T , the vao phitdng trinh thit hai ciia he thay khong thoa

man Vay xet u 0 T i t (*) ta c6

V {u^ - (/) = 1 - uS;" <=> u'^v{l + v'') = 1 + If' ^

K h i 7/.^?; = 1 t a c6 1' = —r, thay vao {v.ny = - 5 t a ditdc

- Cac nghiem ciia h§ 1^ ' ^ f "

•i> ( S I ofiv "/,fii!

B a i t o a n 3 8 he phuang trinh | ^3 :t | ^ 2 ^ ?! 17^ 5 Vi Ui*o

^ 1 2^2 + 4u - 2uv - 4v = Au + 4(; - 2u'^ - 2uv

I uv^ -v + 3uv - 3 = 3uz; + 3 - uv'^ - v

B a i t o a n 42 Giaijie phmng Irinh | ^ ^2) _^ 3.2^2^ ^ 208x2y-

Hvfdng d a n De thay (x; y) = (0; 0) la nghieni cua h§ T i e p tlieo xet xy 0

Ho titdng ditrmg vrJi

r 2x'^y + y^x + 2y 4- x = Qxy

B a i t o a n 45 Gidi he phUdng trinh I J _ + 4_ E = 4

Hu-dng d a n Dat u = x + ^, i ; = y + ^, t a t h u dUdc he | 4("2"^_j_^]2y f

B a i t o a n 48 Gidi he phudng trinh

Bai toan 50 Gidi he phUdng trinh

Giai Bien d6i he da rho, ta tlni ditrJc

LuM y- Dang he ijhitong tiinh giai bang each (hit an phu nay thittJng gap tj

nhieii ky thi, tit DH-CD den thi HSG cap tinh va khu vuc

Bai toan 52 (HSG tinh Ha Tinh, nam hoc 2010-2011) Gidt he

G i a i Xet x = 0 => y = 0 Vay (0;0) la mot nghiem ciia he Xet x ^ 0, chiii

hai ve cua (1) cho x, hai ve ciia (2) cho x^, ta dudc

X + - + y = 3 x2 + ^ + 3y = 5 x''

( x + ^ ) + y = 3

Dat 2 = X + - , ta t h u ditdc he

y\) = ( 4 ; - l ) y;z) = \l-2)

/ f + y = 3

I z' + y = b

k h i (y; z) = (4; - 1 ) : I ^ ; I ^ _ 1 ^ { ^ 4 ^ 0 (vo nghiem)

K h i ( y ; z ) = ( l ; 2 ) , t a c 6 { ; ; ; L 2 ^ { + I = 0 ^ { y = }

Vay he c6 hai nghiem (x; y) = (0; 0), (x; y) = (1; 1)

B a i t o a n 56 Gidi he phuang tfinh | + ^^^2^) = ^

G i a i Do x^ + x/ = I nen ton Lai a e [O; 360"] sao cho x = s i n n , y = coso

Do do (1) trcl tlianh

\/2(sina - co.sa)(l + 2 sin 2a) = \/3

<^v/2.y2sin(rv - 45").2 Q + s i n 2 a ^ = ^3

<^4sin(a - 45")(sin2a + sin30") = \/3 f ^ f ^ I'^i •;

<t=>8sin(rv - 45") sin(a + 15") cos(a - 15") = \/3

<:^4cos(a - 15")[cos60" - cos(2a - 30")] = ^fi fti:^^i*i,3>

«i.2cos(a - 15") - 4 C O S ( Q - 15").COS(2Q - 30") = N/3 ""'^'^ ^.7^;

^ - 2 cos(3a - 45") = ^3 ^ [ I ^^^^^^Q" ^ ^ ' '

Vay lie da clu) C O nghiem v * / »» • m^ i>^' i i f i i - f

(sin65"; cos65"), (sin 185"; cos 185"), (siii305"; cos 305"), (sin85"; cos85"), (sin205"; cos 205"), (sin325"; cos.325") •„ • ^, ,

1.5 Phu'dng phap cong, phifdng phap the - |

Day la phitdng phap cd ban nhat T i t bai hoc "vfl long" ve he phudng trinli

da C O phitdng phap nay Tuy nhien phitdng phap nay van thilclng xuat hien ci

nhrtng ky t h i Idn, nhifng ky t h i chi danh cho nhitng hoc sinh xuat s;lc Sach nao viet vc he phitdng t i i n h cung c6 pluldng phap nay, do vay sau day ta chi trinh bay mot so bai toan kho va di sau lidn vao viec phan tich ky thuat giai cung nhif ky thuat sang tac

V i d u 1 Xuat phdt tit mot bien doi tiMng dudng do ta chon '"^ ,

(x - 2)'* = (y + 3)^ ^ x'^ - 6x^ + 12x = y'* + 9y^ + 27y + 35 (1)

Khi (x; y) = (3; - 2 ) thi (1) dung Cung vdi (x; y) = (3; - 2 ) thi ,

x - ' - y ^ = 35 -nf ,4 (2)

Tii (2) vd (3) /.a c6 bdi toan sau

B a i t o a n 57 Gidt he phuang trinh | l[2f~^yi _ cjy. (2)

G i a i Nhan hai ve ciia (2) vdi - 3 roi cong vdi (1), ta dUdc

x^ -x/ - 6x^ - 9y2 = 35 - 12x + 27y

- 6.r2 + 12.T - 8 = + 9.1/2 + 27?/ + 27 •

^ (x - 2)^ = (y + 3)^ X - 2 = y + 3 ^ X = y + 5

Thay vao (2) ta d U d c »' ~

2 (y + 5)'-^ + = 4 {y + 5) - 9y 5(/^ + 25y + 30 = 0 <^

Nghiem cua he la (x; y) = (3; - 2 ) , {x; y) = (2; - 3 )

Lvfu y. Trong 15i giai tren, quan trong nhat la biet nhan hai ve phitdng trinh

(2) vdi - 3 roi cong vdi phiTdng trinh (1), tai sao phai la so - 3 ma khong phai

la so khac, tai sao lai cong vdi phudng trinh (1) ma lai khong trit ? Dicu bi

mat do nam d "phudng phap gia dinh" (hay con gpi la phudng phap he s6

bat dinh) nhu sau: Xet (1) + Q.(2) ta ditdc , ,

/ + 2ax^ + 3a?/^ = 35 + 4ax - 9ay

T\i (3) ta chon a, a,b sao cho

Di theo hudng nay ta cung tim dUdc Idi giai : Tii (3) ta c6

x^ + 2ax^ - 4ax = i/ - 3ay'^ - 9ay + 35

3 , r,„2

-^x^ + 2ax^ - 4ax - 2'^ = y-^ - 3m/ - 9ay + 3^

Vay ta can chon a sao cho 2nx'^ = -3.x^.2 =^ a = - 3 Qua day thay rang

nhiing he c6 chiia cac hang tiit x^, x^, x va y^, y^, y ta c6 the diing he so bat

dinh de du:a ve cac hSng dang thulc Chu y rang viec xet (l)+a.(2) van khong

giam t6ng quat hdn so vdi viec xet ft.{l) + a.(2), vi khi giai phudng trinh ta

CO quyen chia ca hai ve cho mot so khac 0

V i du 2 Xuat phdt tit mot bien doi tMng diCdng do ta chon

ix-2f =={y + lf -6x^ + 12x = y^-\-3y^ + 3y + 9. (1)

Khi (x; y) = (2; -1) thi (1) dung Cung v6i (x;y) = (2; -1) thi

Ti( (1) va (2) ta dicac 2x2 + ^^2 _ 4^ ^ Q

Ta (2) va (3) ta cd bai todn sau

Bai toan 58 (De thi chinh thiJc Olympic 30/04/2012) Giai he

f x^-y^=9

\2 + y2 - 4x + y = 0

V i du 3 Xuat phdt tii mot bien doi tUOng ductng do ta chon

{x-2)' = {y-At '

^x^ - 8x^ + 24x2 - 32x = y^ - 16y^ + 96y2 - 256y + 240 (1)

Khi (x; y) = (4; 2) thi (1) d,ung, vdi (x; y) = (4; 2) thi x^ - y^ - 240 (2)

Tii (l) va (2) ta ducic

- Sx^ + 24x2 _ ^ _ ; i g y 3 _^ 9gy2 _ 256j^ '

Tii (2) va (3) ta c6 bai todn sau ' ' '

Bai toan 59 ( H S G Quoc gia-2010) Giai he phuang trinh

\3 - 2y3 = 3 (x2 - 4y2) - 4 (x - 8y). {li)

Giai. Nhan phudng trinh (ii) vdi - 8 roi cong vdi phudng trinh (i) ta ditdc

x"* - y4 - 8x3 + 16y^ = 240 - 24x2 + 96y2 + 32x - 256y

•^x'* - Sx^ + 24x2 _ 32^ + 16 = y'' - m/ + 9Q>if - 256y + 256

Vay he da cho chi c6 hai nghiem la (4; 2) va (-4; - 2 )

LuTu y Vi sao lai nhan (ii) vdi - 8 roi cong vdi (i) ? Dicu nay ditdc li giai

titdng tir nhvt luu y d ngay sau Idi giai bai toan 57 d trang 35 Mot dieu lvfu

y nita la mot so thay giao cham thi d ky thi HSG quoc gia nam 2010 cho

bigt, CO rat nhieu thi sinh "bo tay" trudc bai nay, do khong xac dinh dung

plntdng phap, do sii; dung "dao to biia Idn", nhftng cong cu qua manh (chang

ban nhit dao ham), nhflng cong cu manh do khdng thich hdp vdi bai toan

"ay Con ve phan sang tac dc toan, khoug biet ngitdi thiet kc ra bai toan nay

lam nhit the nao ? c6 tildng tit nhit each sang tac da trinh bay d vi du 3 d

trang 37 hay khong ?

N h a i i xet 1 Ddii liieii dO diCa ve Jiang ddiuj tlnic la timi.y kc cd chda

./•^:t //-.,;:•* ± //•',.,:•' ±

V i d u 4 XadI pfiat li! tnoi bihi do/ tUdnf/ dudnij do fa chon

• •• (v + 5f + (a - :if - 0 •(/•* + - 9v'^ + 15o' = - 9 8 - 27u - 75v (1)

f

Kill ((/.: r) - 5 ) //;/ (1) dmui Cfmcj v&i [u: o) •= (3; - 5 ) th'i

""""" = -98 (2)

Dal 11 — i: I // = J- - 1/ {Ill/if ph.rji d/ri hirn, ro' hdii), iJiay vdo (2), (3) d,iM(:

• \- -.h-ii^ = - 4 9

./•- - 8x/y + - 8y - 17x

7);, rci bdi U>uii saa

B a i toan 60 ( H S G Q u o c G i a n a m 2004, bang B ) Cidi lir phiMiui trhili

C a c h 2 Nlian phiWng triiili (2) vdi 3 loi cpng v6i plutciiig tiiiih (1) ta dudc

+ 3xy^ + 49 + 3x'^ - 24xy + [h/ - 2Ay + 51x = 0

<^(,T + l)(x^ + 2x 4- 3?/^ - 24?y + 49) = 0

^ ( : r + l)[(.,; + l)'-^ + 3(y-4)'^] = 0 r + 1 = 0

.r = - 1 va y = 4 r + 1 = 0 va ;// - 4 = 0

L i f u y Cach giai 2 rat ugaii goii va de hiCni, nliitug dang sail su iigan gpii de

hiCu dp ta da phai rat vat va de tim ra IcJi giai nhit sau : D P phirpug triidi

thti' nhiU cua he c6 hac cao uliat iicii ta se de nguyen, nhaii philPng trinh thit

hai cho n roi cpiig vc3i pliitPiig trinh thif iihat ta ditdc : •-'

+ 3xy^ + 49 + n{x^ - 8xy + if + 17x - 8y) = 0 (*)

Mat khac ta c6 thfi nhaiii ditrtc nghiem ciia he la (x; y) = ( - 1 ; 4 ) , do do ta

nioug inuoii (1) phan ti'ch dUdc thauh :

(x + l)(rtx^ + hx + cy^ + dy + 49) = 0 (he so ciia xy trong ngpac bang 0)

<^nx'^ + bx^ + cxy"^ + dxy + 49x + ax^ + bx + ry^ + dy + 49 = 0

<^ax^ + (o + b)x'^ + C X A / + dxy + cy^ + (6 + 49)x + dy + 49 = 0

Tit (*) va (**) doug uhc\ he so ta ditdc :

B a i t o a n 61 Gidi hf pfixMng trinh

Gx^y + 2f + 35 = 0 5x'-^+5y^+2xy + 5 x + 1 3 y = 0 (2) (1)

G i a i Dat suy ra x = y = Thay vao (1) ta duac

Ket luan : He cd hai nghiem la (x; y) = ^ " ^ i " " ^ - (^5 v) = '

Lvfu y V i sao t a lai nhiin (4) vdi 3 roi cong vdi (3) ? Ta cd

Vdi u = 2 ta cd u = - 3 , dan tdi | ^ t j{ ~ 2 <^ (x; y) =

u + A (37i''^ + 977) = - 35 + A ( - 2 7 ; ^ + 47,')

(5)

+ SATi^ + 9A7/ = - 2\v~ + 4At' - 35

Ta can chon A sao cho (5) cd dang

D o n g b a n g B a c b o - 2 0 1 1 ) Gidi he phuang trinh | ^ l ~ ^ 2 I ^ " ^

B a i t o a n 63 Gidi he phuang trinh

Heed hai nghiem (x;y) = (x;y) = ( ^ ; ^ )

-Lifu y V i sao nhan phifdng t r i n h (2) vdi 2 roi cpng vdi phitdng t r i n h (1) ?

Ta union cho ( i ) la n i o l phircJng trinli bac hai theo {3x + y ) , vay t.hi (i) phai

dUdc viet lai thanh

Tir (i) va (ri) d o n g n h a t he so ta duoc

( i i )

"•:m u'

Ta chu y rKng, nhfrng he phitdng t r i n h chvta hang t i t x'^.xy,y'^ phan Idn c6

tho dira vfl phudng t r i n h bar hai theo ax + by

B a i t o a n 64 Gicit he phuang trinh / + +J^y^ + = ^ (1)

Lvfu y Tai sao l a i lay p h U d n g t r i n h (1) c o n g vcii p h i t d n g t r i n h (2) n h a n 2 ?

Y tUcing l a t a se b i e i i d o i de d u a ve p h i t d n g t r i n h bac h a i t h e o m x + ny D6

lam difMi do t a n h a n p h u d n g t r i n h (1) vc'ii a va p h i t d n g t r i n h (2) v d i (3 r o i

D6 cho ddn giaii, ta chon a = \ i3 = 2

Tiep thto ta st phdt tritn them ky thudt yiui dd duoc de cap J each 2 cua Idi

giai bai loan 60 d trang 38 ' '

B a i t o a n 65 Gidi he phUdng trinh

/x4 + 2 ( 3 v / + l ) x 2 + (52/2 + 4 y + l l ) x - y 2 + i 0 y + 2 = 0 (1)

\ t / + (x - 2)2/ + x 2 + X + 2 = 0. (2) • ^ - •

G i a i K h i y = - 1 t i n (2) trci thanh x^ + 3 = 0, v6 nghieni Vay he khong c6

nghiem dang (x; -1), do do c6 the gia si'l y + 1 7^ 0 Nhan phitdng t r i n h (2) cho y + 1, l o i lay phitdng t r i n h (1), t r i t phitdng t r i n h vita nhan ditdc, t a c6 :

(x + y)(x - y + 2 ) ( x 2 - 2x + y2 + 3 y + 5 ) = 0 ^ , \Vdi X = -y tliay vao (2) ditdc (y + 2)(y - 1)^ = 0 <^ y G { - 2 , 1 }

Vdi X = y - 2, thay vao (2) ditdc (y - \ + 4 ) = 0 <^ y £ { - 4 , 1 }

De thay x^ - 2x + y^ + 3 y + 5 = (x - l ) ^ + (y^ + 3 y + 4 ) > 0

Tint lai ta thay he c6 nghiem (x, y) = ( - 1 , 1 ) , (2, - 2 ) , - 4 ) ,

Lu\ y Day la bai toan kho, de co difdc IcJi giai ngfln gon nhit tren t a da

Pliai phan tich, t i m Idi giai n h u sau :

B u d c 1: T i m n g h i e m c u a he Neu biet ditdc nghieni t h i y titdng ciia ta se

ro rang hdn nhifiu Lan lifdt thit x = - 2 , - 1, 0 , 1 2 3, ta tiin dildc 2 nghieni

hoftc ?/ bcii —X (tuy trirdng hop xeni each nao c6 loi), vdi bai nay ta thay

y = —X vao hai phifdng trhih ctia he va t h u clUdc

Viec phan tich tren la khong kho v i t a da biet tiudc nghiem x = - 1 , x = 2

Biidc 4 Li.ta chgn bieu thiJc thi'ch hdp Nhit the, so vdi phitdng trinh t h i i

nhat vita nhan diWc t h i j)hirdng trinh tint hai thieu d i mot bieu thitc la x - 1,

nhung t h u y rftng bieu thiic nay cung tUdng dUdng vdi - y - 1 Ta se chon

mot trong hai bidu thiic nay de nhan vao Ro rang neu chon -y - 1 t h i viec

nhan vcii (2) se tao ra mot da thitc c6 chita y'^ dong bac vdi x'* d phudng

trinh (1) Vay t a se nhan phitdng t r i n h sau cho -y - I

1 t o a n 66 Gidi he phudng trinh | ^ 2 + ^ +J Z {2)

G i a i X e t x = 0, x = 2 ta thay he c6 nghiem (x; y) = (0; 0), (x; y) = (2; 2)

Vdi X ^ {0; 2 } , xet ( l ) [ x ( x - 2)] + (2) ta c6 phitdng trinh

x ( x - 2)(x''^ - 2xy + X + (x"^ - ix^y + 3x^ + y'^) = 0

<^(x - y)(2x''* - x^ + X - 2/) = 0

• K h i X = thay vao (1) : 2x - x^ = 0 Tritdng hdp nay loai do x ^ {0; 2}

• Neu 2x'^ - x^ + X - y = 0 ta CO he phitdng trinh

2x^ - x^ + X - ?/ = 0 x^ - Ixy + X + y = 0

Cong hai phitdng t r i n h t a c6

2x^ - 2xy + 2x = 0 2x(x^ + 1 - y) = 0 x^ + 1 = y (do x 7^ 0)

Thay vao (1) t a c o

- 2x(x2 + l ) + x + x2 + l = 0 - » - 2 x ^ + 2x2 - X + 1 = 0 <^ X = 1

Ttt do t i m ditdc (x; y) = ( 1 ; 2)

Vay he da cho c6 nghiem (x; y) = (1; 2), (0; 0), (2; 2)

Lvhi y L d i giai tren dildc t i u i ra nhil sau : Dau ticn t a thay

( l ) < i ^ x 2 - 2 x y + x + y = 0

Co the t i m dUdc 3 nghiem nguyen ciia he nay la (x; y) = (0,0), (2; 2), (1; 2)

•2x = 0

- 2 ) p = 0

Tit 2 nghiem dau t a thay x = y Thay vao he t a dudc ^ % M 2 ^ _

Nhit vay ta se phai nhan x{x - 2) hoax- y(y - 2) vao (1)

jyici r o n g Nlnt <ta de cap, n e u biet cang nhieu nghiem t h i ta c6 Idi giai cang (iop Sau day l a mot cacli phfui tich khi t a biot c a 3 nghiem Ta so Ian litdt,

(y = X

lap 3 quan he tuyen Mnh gnla x va y Trong bai nay, do la | y = 2x Do (2)

CO bac ca.o luJn nen ta xet / ^iiunj >s!stvijii<'( ,<«,>•

thu d i t d c 2;;;'' - x~ + x - y = 0 Vay ta di den Idi gitii ng;ln gon nhu s a u :

Ldi g i a i Thtt vdi x G A/ = {0; 2: 1; 2} ta t i m d U d c nghiem

Cong lai t a cd 4,r-* + 2x = tJ ( s a l do x ^ M). Vay I r u - d n g hdj) nay loai '

Tom lai he cd nghiem {x; y) = (t); {)) (2: 2) (1; 2) •

Tirdng t u , t a cung c6 y = 2 Vay he da cho c6 nghiem la (x, y) = (2, 2)

Lvtu y Ngoai each giai tan dung t i n h chat cua cac can thuTc, t a ciing c6 the

dat an phu roi bien d6i Trong phitdng trinh t h i i hai, cac so hang tit do c6

the khac nhau ma Idi giai van dUdc tien hanh tUdng t u Chang han, giai he

G i a i De thay rang (x; 0; 0), (0; y\ (0; 0; z) thoa man ho da cho Tiop thoo

xet xyz ^ 0 K h i do chia cac phUdng t r i n h cua he cho x^j/^z^, t a dUdc

B a i t o a n 7 0 ( C h o n d o i t u y e n K h a n h H o a , n a m h o c 2 0 1 0 - 2 0 1 1 ) Gidi

(1) ' (2)

Phirong trinh tlii't hai cua he tiTdiig ditdng vdi

G i a i Dicu k i e u : x > 0,?/ 7^ 0 Phittliig t r i n l i t l i i i u h a t cua he tifOiig dudug

y ^ x + y'^ 2xy/x + 2xy<^y^ + y (v^x - 2x) - 2 x v ^ = 0

nghich bien tren moi khoang

> 0 nen / dong bien, (/ {x) = —

he da cho c6 nghiem dviy nhat la (x, y) = (\/3,2v/3)

LtTu y Quan he cua x va y dUdc che giau ngay trong phitdng trinh dau tien,

neu nhan thay dieu do t h i cac bitdc tiep theo se de nhan biet Bai nay tinh toan tuy rifcJm ra nhimg hirdng giai rat ro rang nen khong qua kho M o t dicu

(iang quan tarn nfra la do ham g nen bat buoc ta phai xet tren hai khoang

B a i t o a n 78 ( H S G t i n h L a m D o n g , n a m hoc 2010-2011) Gidi he

.i.'v ••••'

Hu-dng d a n Lay (1) trit (2), ve theo ve, ta d\Mc :

y4 - 2y2 - 4xy3 + 4xy + 1 = 0 (y2 - 1)^ - 4xy (y2 - 1) = 0

^ ( y ^ - l ) ( y 2 - l - 4 x y ) = 0

«>y = 1 hoac y = -1 hoac y^ - 1 + 4xy = 0 h V- ;•

• N€u y = 1, thay vao phildng trinh (1), ta d\rdc : •

4x2 + 1 - 4x = 1 <^ X (x - 1) = 0 X = 0 hoac X = 1

Neil y = - 1 , thay vao phiTcfng tnnh (1), ta ditdc -x"'

L u ^ y Day la m o t dang he pliifdng t r i n h da thite k h a k h o , 10 rang neu

phifdng t r i n h t h i t hai, ngitdi t a chia hai ve cho 2 t h i kho c6 the nhan biet gia

t r i nay ma n h a n vao r o i t r i t tiJfng ve nhif tren Viec phat hien ra gia t r i 2 dg

nhan vao c6 the d i u i g each dat t h a m so p h u r o i hra chon ^

B a i ai t o a n 79 Giai he phuang tnnh | ^ ^J^y

+ x^ = 1 8 v ^

G i a i Ta Ijien doi tUdng ditdng | ^ ^^2^^ 18^2 (2) ^'^

la nghiem t h i y > x > 0 Tijt (2), r u t y theo x va the vao (1) co

Chi'mg to h a m / dong bien t r e n (0; +00), nen (4) co nghiem t h i nghiem do duy

nhat T i t do he co nghiem (xo; yo) t h i nghiem do la d u y nhat De t h a y y = 2x

t h i i\i (1), t a co: x^ = 4 hay x = \ / 2 Suy ra y = 2v/2 T h i i lai (v/2; 2^/2) thoa

man (2) T o m l a i he da cho co nghiem duy nhat la (x; y) = (\/2; 2y/2)

Tiep theo, ta xet w.ot phiidng phdp sang tdc di toan cung rat dan gian nhxtng

duac svC dung nhieu trong thdi gian gan day, do la xuat phat tit mot he da

, it , su ^at gtdi, Chung ta thay thi hmh tMlc cua cdc hif,n co mat trong he vd

f t dSi rut gon ta thu duac mot he c6 hinh thvCc hoan toan xa la vdi cai he

Bai loan 80 Gtdi he phuang tnnh | 5^4 _ 4^6 ^ ^ 2 (2)

G i a i Neu x = - 1 t h i thay vao (1) thay vo l i G i a sit x ^ - 1 T i r (1) suy ra

Hay X G |o, 1, Vay he da cho co 3 nghiem: (0; 0), ( 1 ; 1), ^ \2'2)

Lvtu y Cach giai nav c6 m o t m di6m la khong can phai m a n h khoc gi ca

ma chi can bien d o i het site b i n h thitdng Ngoai ra con co each giai khac, do

la di ngUdr lai qua t r i n h sang tac dc toan _

V i d u 6. Tit he I ^2"^ ^ t V ^ (^''^" y cd nghiem ( 1 ; 2 ) ) Thay thi

x hai ~ vd y bdi y^ tin ta co he : 2x-^ -•

2x3 2x3

Vdy ta CO bdi toan sau

1.6 Phifdng phap dao an, hang so bien thien

1.6.1 P h i f d n g p h a p g i a i mi

Xet phUdng t r i n h an x, tham so 711 : f { x ; in) = 0 Tuy nhien k h i giai ta lai

coi an la 771, tham so la x Giai 771 theo x roi quay lai an x Ta thirdng diing

phitdng phap nay khi tham so m c6 mat vdi bac hai va biet thiic A ciia

phitdng trinh l)ac hai an m do la bieu tliifc chinh pliUdng (A la binh phitdng

ciia mot bi6u thi'tc) Trong mot so tritdng hdp ta con c6 the coi so la an, day

la rapt phitdng phap rat dac biet : Phitdng phap hftng so bien thien

B a i t o a n 82 (Phifdng p h a p dao a n ) Tim m d i phUdng trinh sau c6

Lifu y. Ta cd th? t i m ditdc 771 theo x bang each van dung dinh If V i - ct dao

nhit sau : Phitdng trinh ( 2 ) cd 5 = va

• Thay a = vT7 vao (2), t a ditdc \ / T 7 = - x 2 Phitdng trinh nay v6 nghiem

• Thay a = s/Vf vao (2), ta ditdc

yrr = l ± ^ ^ x ^ - v T 7 x 2 + 2 = o ^ x 2 = ^

a 7 ± 3 Phitdng trinh da cho cd 4 nghiem ± ^ ^

Lvtu y. Doi vdi bai toan nay, ta cd the dat an phi.i dita ve phitdng trinh bac

ba, ma ta biet r i n g phUdng trinh da thiic bac ba luon giai dUdc, nen phitdng phap giai nay luon thanh cong Tuy nhien thay ngay rang giai theo phitdng phap tham bien la gon, dep lidn, rat doc dao

B M toan 84 Giai phudng trinh log| x + log2 ^ = 5 log^ 8 + 25 log^ 2 (1)

G i a i Dieu kien 0 < x 7^ 1 Dat log2 x = « K h i do

" 2 ^ = 2 ^ ^ = 2 ^ , thoa dieu kien

Nghiem cua phitrfiig t r i n h da cho la x = 2 ^ ^ , x = 2 " ^ ^

Phirdng t r i n h c6 hai nghiem x = x =

B a i t o a n 8 6 Giai jj/it/ofnp iHn/i a 2 " i 3 4 = 201^3/^^-^-;^:

(1)

^^^^^ G i a i T a coi (1) la phirong trinh an a Ta c6

(1) ^ a 2 ' » 3 ^ 2 0 1 ^ ; ^ - X ^ fl = - ^ - „

<=> a = / (/(a)), v6i /(a) = ^ " ^ / ^ ^ ; , , 7 ! l A J (2)

Ham s6 / biSn a c6 /'(a) = ^ o i g ^ " 1 ^ ( ^ 7 3 ^ ^ > " ^ ' 1 " ' ' " " ^ bifin tren E

NSU a > /(a) t h i /(a) > / (/(a)) ' " U " /(a) > a ^ a = /(a), \

N i u a < /(a) t h i /(a) < / (/(a)) "^'4'' / ( « ) < a ^ a = /(a) 'I

Tom lai, do / doug bien ncn ,

(2) ^ a = /(a) 4 ^ a = - ^ " ' ^ / ^ T ^ ^ a 2 " ' 3 = a - x ^ x = a- a^^''\

Phitdng t r i n h (1) c6 nghieni duy nhat x = u a 2 " ' ' ^

L i f u y Ban doc hay lien he bai toan nay vdi Ijai toan 93 6 trang 223

1.6.2 P h i f d n g p h a p s a n g t a c b a i t o a n m d i

V i d u 1 Bang each khai trien phUdng trinli tick ' '' i c ;

(x2 - 3 x - 1 - a ) ( x 2 - 5 x - a ) = 0 ' ''^

ta duuc bai loan sau .' ^^'^

B a i t o a n 8 7 Gidi phiCcJng trinh

Phuong t r i n h (3) va ( 4 ) Ihu lifdt c6 biet thirc ' S

A i = 9 + 4(1 + « ) = 13 + 4a, A 2 = 25 + 4a swsi

1;-• Neu A i < 0 hay a < - — thi (3) v6 nghiem

• Neu A i > 0 hay a>-— t h i (3) <^ a; = x i , 2 = •

Ta CO bdi todn sau

B a i toan 88 Gidi vd bien ludn phUdng trinh sau theo tham so a

- lOx^ - 2(a - ll)x''^ + 2(5a + 6)x + 2a + = 0

To CO bdi todn sau

B a i toan 89 Gidi phuong trinh

X'^ X

, 2; la nghiem ciia phirdng trinh (1) khi va chi khi j

• Thay a = VTO vao (3) ta dudc VT9 = -2x^ PhUdng trinh nay vo nghiem

• Thay a - \ / i 9 vao (4) ta ditdc

X^ J ;

Pliitdng trinh da cho co 2 nghiem ± y ' ^ ^ ^ - ^ — •

a.=e- 3t, 0 2 = - t ^ + 5 i ^ { 2 ; + ^ = I V _ 15,^;;! ^ Vat/ tti t^d 02 /d nghiem cua phxtOng trinh dn a sau

logl X - 8 log3 X + 15 = 49 log2 3 - 14 log^ 3

O log3 X - 8 log3 X + log3 9 + 1 3 - 49 log2 3 - 7.2 log^ 3

^ logj^ X - 8 loga + 13 = 49 log2 3 - 7 log^ 9

v/3

B a i toan 90 Gtdi pincang trinh

^ ^, , ^ , l o g ! X - 8 log3 ^ + 13 = 49 log2 3 - 7 log, 9 ' (1)

G i a i Di^u kieii 0 < x 7^ 1 Dat log;, x = t K h i do

log.^ 3 = - , l o g , 9 = - , log3 = log;i X - Iog3 34 = < - -

Tliay vao i)luWiig tiiiili (1) t a ditcJc

V? (2 - 8i + 2 + 13 = ^ - — ^ i"* - + Voi^ + 14i - 49 = 0 (2)

Dat a = 7, phUdng trinh (2) trd thanh

Vrti t = , ta ditdc logy x = x = 3 2 , thoa dieu kien

Nghicui ciia phudng t i i n h da cho l a x = 3 ^ 2 , 2 ; = 3 2

Tiep tuc chon a = v/x^ — x + 1, ta dUc)c

- X + 1 - (x^ + 2) V x 2 - x + T + x^ + x^ - 2x = 0

•»\/x2 - X + 1 =

Ta dUdc bai todn sau

+ 2x^ - 3x + 1 x2 + 2

C a c h 1 Tudng t u nhir bai toan 91

C a c h 2 PhUdng trinh tiWng dirong

\/x2 - X + 1 = X + 2 - % ± | ^ V ^ ^ ^ ^ - ( x + 2) = - ^ (i)

x2 + 2 Neu \/x2 - x + 1 + (x + 2) = 0 t h i

Lifu y. Ta c6 mot each nhanh hdn de sang tac ra nhiing bai toan c6 de bai

va idi giai tirong t i t nhir bai toan 92, do la bien d6i phudng t r i n h tich

\/x2 - x + 1 - (x^ - x ) l [ v / x 2 - x + l - (x + 2) = 0

^ 3 , 2 _ + 1 _ (x2 - X + X + 2) x/x2 - X + 1 + (x2 - x) (x + 2) = 0

^ 3 , 2 _ ^ + 1 _ ( ^ 2 ^ 2) \/x2 - x + 1 + x^ + x^ - 2x = 0 ' |, '

^ (x2 + 2) v/-'c2 - X + 1 = x^ + 2x'^ - 3x + 1

phifdng pliap n^an va chia cho bieu thitc hen hdp (nhit 16i giai theo each 2)

se dtfdf t r i n h bay chi tiet d muc 3.6.5 : Sang tac phUdng trinh v6 t i tiJT cac nghieni chon s&n va phUdng phap nhan lUdng lien hdp (d trang 204)

]^.7 Phu'dng phap s i i dung dinh li Lagrange

Day la "lOt phildng phap dac biet, mdi xnat hien trong thfJi gian gan day

Cd sd ciia phvrdng phap nay la dinh l i sau

D i n h ly 1 ( D i n h li L a g r a n g e ) Neu ham s6 y = / ( x ) lien tuc tren down

[a; h\ c6 dao ham tren khodng (a; b) thi ton tai mot so c & (a; 6) sao cho:

/ ( 6 ) - / ( a ) - / ' ( c ) ( 6 - a ) Sau day ta se trinh bay mot vai dang phUdng trinh dUdc giai bang each van

dung dinh li 1 Tuy nhien con nhieu dang khac nita chita ditdc de cap den, de

giai chiing ta can nhan ra dac thii ciia dinh l i Lagrange dUde chc dau trong phiTdng trinh >

Dang 1 PhiTdng t r i n h a''(=") - b'^^^^ = k{a - b)h{x) , vdi 0 < a ^ 1,

Ttt day t i n i dUdc x, sau do thit lai dd chon nghiem

^ a i toan 93 ( D e nghi O l y m p i c 3 0 / 0 4 / 2 0 1 0 ) Giai phuang trinh

3™'^-2^°=^ = cosX (1)

G i a i De thay x = 0 la mot nghiem cua (1) Gia si't a la mot nghiem bat 1^]

cua (1) K h i do S'^"^" - 2™«" = cosa ^ 3™ *" - 3cosa = 2'^"«« - 2cosa (2)

Xet ham so f{t) = - i c o s a , v6i t > 1 Ham so / lien tnc tren (1; + ( X )

va CO f'{t) = co^af°^'^-^ - cosa T i t (2) ta c6 / ( 2 ) = / ( 3 ) Ham / hen t i , ,

tren doan [2; 3] va c6 dao ham tren khoang (2; 3), do do t6n tai h e (2; 3) sao

Nghiem cua (1) la x = - + k-K, x = kin (k 6 Z)

B a i t o a n 94. Gzdi phuang trinh x'"^^'' = 2'''83-^ -|- loggx^

G i a i Dicu kicn x > 0 Gia sii phudng trinh co nghiem x = a, tiJc la

^ -o)^> ^ i o g 3 7 ^ 2 i " g 3 " + iog3 Q5 ^ y i o g , " _ 2'°e3" = (7 - 2) loga a

^7log3" _ 7 iog3 a = 2'"&i« _ 2 a

Xet ham so f{t) = i ' ° g 3 « _ t logg a, vdi t > 0 K h i do tfr (2) t a co

/ ( 7 ) = / ( 2 ) ^ / ( 7 ) - / ( 2 ) = 0

Ham so / lion tnc tren doan [2; 7] va co dao ham

fit) = (log3 a ) / ' ° S 3 _ iog3 a = - l ) log^ a

Theo dinh l i Lagrange, ton tai c e (2; 7) sao cho

/ ( 7 ) - / ( 2 ) = / ' ( c ) ( 7 - 2 ) = > / ' ( r ) = 0, ' nghia la

Thay x = 1, x = 3 vao (1) thay thoa man Vay tap nghiem cija (1) la { 1 , 3}

B a i t o a n 95 Giai phuang trinh 7™'^ - n™'^' = 1 2 c o t x

Hifdng d a n Dicu kicn x ^ kn, k G Z Gia sii a la mot nghiem ciia phircfng

P h i f d n g p h a p Xet ham s6 bign t : fit) = {t + df^''^ - t'^^^l T i f phitdng

„h da cho t a co / ( a ) = f{b) ^ / ( a ) - /(f)) = 0 Thco dinh h Lagrange, ton

12 c G (b; a) sao cho fib) - fia) = / ' ( c ) ( 6 - a) =^ / ' ( c ) = 0 T i t day t i m dUdc

3 sau do thiit lai dd chgn nghiem

1 1

gai t o a n 96 Gidt phuang trinh — - — =

Giai. Tap xac dinh M Phudng trinh da cho viet lai

Giai. Dieu kien x > 0 G i a siir a la nghiem ciia phuong trinh (*) K h i do

Xet ham s6 / ( O = {t + 3)'°^^" - vdi t > 0 K h i do t i t (1) ta c6

Thay x = 1, x = 5 vao (*) thay dting Vay 5 = { 1 , 5} la tap nghiem ciia (*)

Bai toan 98 Gidi phuang tnnh 3^ (4^ + 6^ + 9^) = 25^ + 2.16^

Giai. Goi a la mot nghiem ciia phitdng trinh da cho K h i do

Khi a > 1 ' ham so g{t) = t " " ^ dong bien tren (0; +oo) suy ra

( r + 1 3 ) " i + (r + 4 ) " i > c " i + (c + 2 ) " i ^ j p ' ^ vay a > 1 khong thoa man (2)

-, k h i a < 1-, ham so ^(O = ' " " ^ nghich bien tren (0; +oo)-, sviy ra

(c + 1 3 ) " - ' + (c + 4 ) " - i < c " - i + (c + 2 ) " - ' • ^' Vay a < 1 khong thoa man (2) ' '

• De thay o = 1 thoa man (2)

V£y (2) <^ « = 1 Do do /'(c) = 0 k h i va chi k h i a = 0, = 1 Thay

-r = 0, X = 1 vao phiTdng t r i n h da cho thay thoa man Vay x = 0, x = 11a cac

nghiem ciia piiitdng trinh da cho

Sau day la mot so dang phrfdng trinh khac diTdc giai bang each dya tren dac thu ciia dinh If Lagrange

Bai toan 99 Giai phUdng tnnh 2^ - 2-'''+^ = (x + 1) 3^

Giai. Tap xac dinh K Xet ham so /(x) = T xac dinh va lien tuc Ircn R, c6

/'(x) = 2"" in 2, Vx e K Theo dinh l i Lagr-ange, c6 c nam gifra x va 2x + 1 :

2 X _ 2 2 X +1 ^ y(^) _ |(2x + 1) = /'(c) [x - (2x + 1)] = - (x + 1) 2' h i 2

Bcii vay phUdng t r i n h tUdng diTdng ( x + l ) 3 - ^ = - ( x + l ) 2 ' ^ l n 2 ^ ( x + l ) ( 3 ^ + 2 ' ^ l n 2 ) = 0 < ! = ^ x - - 1 PhUdng trinh da cho c6 nghiem duy nhat x = - 1

Bai toan 100 ChtCng minh rang phuang trinh

5x2 - 7x + 2

( 1 0 x - 7 ) l o g „ x + - ^ : 3 - ^ = 0 (1)

luon CO it nhat,.mdt nghiem x G -;1

\<5 / Giai. X e t F{x) = (Sx^ - 7x + 2) logn x, xac dinh, lien tuc tren

B a i t o a n 1 0 1 Cho phuang trinh a„a;" + a„_ia:"-i + • + aix + ao = 0.(1)

Chiing minh rang neu ton tai m e N* sao cho

,jn '„ " + "^ n + m - 1 m + 1 m ~ ^'

phuang trinh (1) /won cd nghiem x G (0; 1)

G i a i Xet bam so xac dinh va lien tue trcn doan [0; 1]:

n + m n + m - 1 m + 1 m '

Ta CO

= x"*-^ ( a „ x " + a„_ix"-^ + + aja; + Tren khoang (0;1), t a c o

(1) ^ x"*-! ( a „ x " + a „ _ i x " - i + + aja; + = 0 <^ F ' ( x ) = 0 (3)

L^i CO

7t + m 7i + r« - 1 ^ ^ , n + 1 m ' ~

Theo dinh h' Lagrange, c6 XQ G (0; 1) sao cho F'(xo) = ^ ( ^ j ~ ^(Q) = 0 Do

do tu: (3) suy ra dieu phai chiing minh

B a i t o a n 1 0 2 Giai bat phuong tnnh 3^'-'' + (x^ - 4) 3^-2 > i

G i a i Xet ham so / ( x ) = 3^ xac dinh va lien tiic tren R, c6 /(O) = 1 va

f'{x) = 3^ In 3, Vx G R Theo dinh If Lagrange, c6 c n l m giiJa 0 va x^ - 4 sao

cho : / (x2 - 4) - /(O) = /'(c) (x2 - 4 - 0) B i t phudng t r i n h da cho v i l t lai

/ {x^ - 4) - /(O) + - 4) 3^-2 > 0

^ / ' ( c ) (x2 - 4) + (x2 - 4) 3^-2 > 0

<^3'=ln3 (x2 - 4) + ( x 2 - 4) 3^-2 > 0 (x2 - 4) (3<^ln3 + 3^-2) > 0

^ x 2 - 4 > 0 ^ X e (-ex; - 2 ] U [2; + o o )

L i f t ! y M o t \'ai he lap ba an cung c6 the dudc giai bang each van dung dinh

li Lagrange (xem bai toan'99 d trang 284)

^ g P h i i d n g p h a p h i n h h o c

I g I P h e p d o i b i e n du'dc d i n h h t f d n g b d i p h u f d n g t r i n h t h a m

s6 c u a d i f c t n g t h S n g « j • s i r

Svf dung ket qua rat quen thupc : Neu dudng thang A di qua digm M ( x o ; yo)

va CO vectd chi phitdng li = {a; b) t h i A c6 phUdng t r i n h t h a m so:

{i = yo + bt ( ^ e R )

r-Ket qua v6 cimg ddn gian tren lai cho ta mot phep dat i n phu rat dep cQng iihu nipt each sang tiic dc toan rat nhanh chong

V i d u 1 Trong mat ph&ng Oxy, xet diidng thang A di qua diem iV/(l; 3) vd

CO vecta phap tuyen li = (1; 3) Khi do A cd phuang trinh tham so vd phuang

trinh tSng qudt Ian luat Id

Trong (3), thay u bdi x duac Vx^ + 8 + 3 \ / l 2 - x^ = 10 Ta cd bai toan sau

B a i t o a n 1 0 3 Giai phuang trinh Vx^TS + 3^12 - = 10 ^ (*)

G i a i Di^u kien ( f + ^ > 0 ^ -2 < x < VU Dat = I + 3t

^Sy V x 3 + 8 = 4 < ^ x ^ + 8=16<!=»x^ = 8 < ^ x = 2 (thoa man dieu kien)

y B i quyet nao dan den phep dat V^^TS = 1 + 3 i ? Do chmh la phitdng

^''inh tham so cua ducing thang A : Doi vdi bai toan tren c6 c6 each giai khar,

^0 la dat p = v / ^ 3 T 8 , q = s/U^, dua v^ he { ^ 2 ^ 9 2 =^20

j,n; ^ ^Ix + 6 = 5i - 2 \x + 6 = 125C,^ - ISOf^ + gOt - 8 (3)

Thay (2) vao (3) ta diiOc

- 1 = 125i^ - 178*2 + 60i - 8 44 125i^ - ITSi^ + mt-7 = Q^t = l

Vay \/x + 1 = 2<;^x + l = 4 - ^ x = 3 (thoa man dieu kien)

Lvfu y. Phep dat yjx + 1 = 2i dudc t i m ra nhu sau Ta xem ^/x + 1 = X ,

v^7x + 6 = Y K h i do tu: ( 1 ) c6 -2Y = 4 Trong mat phring vdi he toa

do OXY t h i bX 2 y = 4 la phiWng t i i n h ciia diroiig tliang A qua (hcni

A/(0; - 2 ) va c6 vectd chi phUdng "u = (2; 5) Vay A c6 phitdng trinh thani

so I Y ="^'2 + 5 i ^^^^^ S*?* y '^'^^ V x T T = 2t

B a i t o a n 105 (Phufdng p h a p h a m lien t u c ) Giai bat phuang trinh

V x + 1 + -^7 - X > 2

G i a i Dieu kien - 1 < x < 7 Xet phudng trinh \/x + 1 + = 2 (*)

Dat \/x + 1 = 1 - /., dieu kien t < 1 Ket hdp vdi (*), dudc

Vdi * = 1 ta CO X = - 1 , vdi i = - 2 ta c6 x = 8, vdi t = - 3 ta c6 x = 15

T h i i lai thay x = - 1 , x = 8, x = 15 thoa {*) Do do (*) c6 ba ughieni

x = - L x = 8,x = 15 V i / ( x ) = v / x T T + ^ 7 - X - 2 la ham so lien tuc tren

nijfa khoang [ - l ; o o ) nen / chi doi dan khi - i , , ^ - ^ - \ ^ s x ' ^ ^ - ^

d i qua cac diem X = - l , x = 8,x = 15 Ta

CO / ( 7 ) = v / 8 - 2 > ' 0 , / ( 9 ) = y i 0 + </=:2-2 < 0 , / ( 3 4 ) ^ > 0

w p o do ugliiSm ciia bat plutdng trinh la - 1 < x < 8, x > 15

L t m y- Giai bat phitdng trinh b i n g "phitdng phap ham hen tuc" nhiT bai toan 105 nay con dUdc de cap d bai toan 4, trang 159, 6 bai toan 77, trang

209 va bai toan 78, trang 210

B a i t o a n 106 Gidi he phuong trinh | = 4 ( 2 )

G i a i Di6u kien { ^y->"~o.' ^ " ~^ Dat V ^ T T = 2 + t Thay vao (2), ta

dUdc v / y T T = 2 - i v a dieu kien ciia * l a - 2 < t < 2 K h i do

Ket hdp vdi dieu kien ditdc * = 0 Vay (x; y) = (3; 3) (thoa man dieu kien)

1.8.2 M o t so phu-Ong t r i n h , h e phifdng t r i n h , b a t phu-dng t r i n h

V i d u 3 Ta se xay dUng mot phUdng trinh dua tren ket qva :

•^."6 = |"a I | V | <^ cos ^"a , "6 ^ = 1

< ^ ( " o , " ? ) = 0 « > 3 A ; > 0 : "a = fcV

Xet ^ = (x; 1) , V = ( y 3 F + 2 ; ^ 4 ^ Khi do

1 ,

^ V = x V S ^ + \ / 4 ^ , 1^1.16 1 = \ / 2 ( x 2 + l ) ( a ; + 3 )

% -^.~b = I"?! |V| nghla la x^i^H^+V^T^ = y/2 {x^ + 1) {x + 3 j Ta

ducfc bai toan sau

B a i toan 107 (Dg nghi Olympic 30/04/2010) Gidi phuang trinh

Vay (1) CO nghia la ~a/b = \~a\.\h \^ 7? = fcV, A; > 0 Do do {x]y) la

nghieiu cua (1) klii va chi khi t6n tai k>0 sao cho

Tap nghieni ciia (1) la {2; 1 + ^/2]

V i du 4 Ta se xay dmg mot bat phudng trhih dua tren kit qua :

u.'v > \'u\.\'v \V = \ \ It = k'v

Xet u = {^x - 1; X - 3 ) , 7 = (1; 1) Khi do

'u.'v = V i ' - 1 + X - 3, \ ^ 2 ( x- 3) V 2 x — 2

Vay lt.lt > \lt\.\lt\ la VT^+x-S > ^ 2 {x-3f + 2x~2 Ta duac

bai toan sau

B a i toan 108 Gidi bat phuang trinh y/F^ + x - 3 > \/2x^'^nmfTJ6

G i a i Difiu kicn | ^ g > ^ ^ x > 1 Bkt phirong trinh vigt lai

Bat phUdng trinh co nghiem duy nhat x = 5

j3ai toan 109 (De nghi Olympic 30/04/2010) Gidi he phuang trinh

V i d u 5 DiXa tren kit qua : "Mat phdng ( P ) tiep xuc vdi mat cdu (S) khi

vd chi khi d{I, (P)) = R, vdi I vd R Idn luat la tdm vd ban kinh cua [S)", ta

CO thi xdy dung d.U0c mot so he phUdng trinh Chang han chon mat cdu tdm

0 ( 0 ; 0; 0), bdn kinh R = 1, c6 phucing trinh (5) : + y"^ -\- = I Dieu kien

Ta CO bdi todn sau

B a i t o a n 111 Gini he phudng trinh

jx^ + y^ + z^ = l (1)

\ + 2z + 3 = 0. (2)

G i a i Trong khong gian vdi lie true toa do Oxyz, t a c6 (1) la phifdng trinh

ciia mat cau ( 5 ) tani 0 ( 0 ; 0; 0), ban kinh R=l (2) la phirdng trinh cua mat

p h i n g t a ggi la ( P ) V i d{0; (P)) = 1 = R iien ( P ) tiep xuc vdi mat cau (S)

Vay he tren c6 nghiem duy nhat, nghieni ciia he la toa dp tiep diem ciia ( P )

va (S) Gpi d la ditdng thang qua 0 ( 0 ; 0;0) va vuong goc vdi ( P ) PhUdng

trinh ciia (d) la

r X = 21

\ = 2t ( i e Xet he

Tiep diem A/ (^ ; - ; - - J He c6 nghieni duy nhat (^^^ 2/5 2 ) = f " g ! 3! - 3 j •

B a i toan 112 Giai he phUdng trinh

{ 20072-^""« + 20 082/2009 + 2OO922010 ^ 2OO8 (1)

{ a:2 + 2;2 + 22 + 2a: + 4j/ + 6 r - 7 = 0 (2)

Hu'ding dan Trong khong gian vdi he toa dp Oxyz, t a c6 (2) la phiTdng

trinh mat cau ( 5 ) , tani / ( - I ; - 2 ; - 3 ) , ban kinh R = \f2\ (3) la phudng

nen

t | ! h mat phang (Q) : 2x + y + 4^ - 5 = 0 V i d{I (Q)) == ^ = ^ '

(Q) tiep xuc vdi mat cau ( 5 ) , do do tCr (3) va (2) ta dUdc | 2/ = - 1 Thay

/ ^ ^ i l vao (1) thay thoa man Vay he phUdng t r i n h c6 nghiem duy nhat

g a l toan 113 Tim rn de' he phUdng trinh sau co •nghiem (Utu

< 3.2 + y2 ^ 2^ - 2rnx + m2 + m - 1 = 0 ' N ' V )

\ + v/ + 22 + m - 2 = 0 ,,,,, :~

• rs • Giai. He viet lai | i, + ^ V 2z + rn - 2 = 0, (2)

^ j^g^j 1 _ „ , < { ) <^ m > 1 t h i tif (1) suy ra he v6 nghiem, m > 1 khong thoa

• Neu 1 - 771 = 0 <^ 771 = 1, thay vao he t a c6 , /

^ r {x^i)\+y' + zi=o ^ifjzl

He C O nghieni, vay m = 1 thoa man

• K h i 1 - ni > 0 <^ III < I, trong khong gian vdi he toa dp Oxyz, (1) la

phifdng trinh ciia mat cau (SVn) tam 7(7ri;0;0), ban kinh R = \ / l - m (2) la phUdng trinh ciia mat phang (P) •.x + y + 2z + ni - 2 = 0 He c6 nghiem khi

va chi k h i (5m) va mat phang (P) c6 diem chung, tilc la

• d(I, (P)) <R^ < - 777

v6

<=>27«-^ - m - 1 < 0 <^ - - < 7 / i < l

Ket hdp lai, t,a c6 7 r i G thi he C O nghieni

B a i toan 114 Tim m dr' he phur/ng trinh sau co nghiem duy nhat