2.5.1 M o t s o d a n g p h i f d n g t r i n h p h a n t h i J c h i h i t i . - v, a?x^ > - \\
D a n g 1. PhU'dng t r i n h d a n g + -, = b.
Phifolng p h a p . T a c6 (*) tifdng dirdng ' ' ax
x —
^ ax . / X \ X 0 . 9
+ 2x. = 6<f=> + 2 a . — + = 6 + a^
(x + a)J x + a \x + aj x + a St ly = . G i a i phitdng t r i n h bac hai theo de t i n i x. x2
x + a • ' ,, '
B a i t o a n 5 8 . Giai phMng trinh x'^ + ( = 1. (1)
V x - i y
G i a i . D i c u k i c n x ^ 1. D o a^ + h'^ = [a + bf - 2nb nen (1) tUdug ditdng
2 / 2 \ X , I x'^ \ X
, , ^ V - 2 ^ = : ô ( ^ V - 2 ^ = i. (2)
x - 1 V a : - 1 / x - 1
X - 1
Dat t =
X - 1 . Thay vao (2) ta dUdc t'^ - 2t - I = 0 ^
t = \ sf2.
• Vai 1 + v/2 ta (•6 = 1 + v/2 <=> x 2 - (1 + sf2)x + 1 + ^ 2 = 0. (3)
X - 1 ^ '
V i A i = (1 + sf2f - 4(1 + \/2) = - 1 - 2v/2 < 0 nen (2) v6 nghiem.
• Vdi i = 1 - \/2 t a CO = \-^f2^x^-{\- \/2)x + 1 - ^ 2 = 0. (4) x^
X — 1
Bdi v i A2 = (1 - - 4(1 - \/2) = 2^2 - 1 > 0 nen (4) c6 n g h i f m la 1 ~ \ / 2 i \/2\/2 - 1
X =
\,
Vay (1) CO hai nghiem la x = 1 - y 2 ± V 2 7 2 ^
B a i t o a n 59. Gidi phuang trinh x^ + 9x2 (x + 3)2 G i a i . Dieu kien: x ^ - 3 . K h i do (1) tuong dirong
= 7. (1)
X — 3x \ V~ x + 3y
,2
+ 6. X
x + 3 = 7 ^ 2 \
' ^ + 6 .
x + 3y x + 3 = 7.
Dat ij = k h i do + _ 7 = 0 4=> " ^ x + 3
K h i (/ = 1, t a CO x'^ = x + 3 x =
y = - 7 . 1 ±
i'U
• K h i y = - 7 , t a co phiWng t r i n h x^ + 7x + 21 = 0 (v6 nghiem).
Vay phudng t r i n h (1) co tap nghiem: S = |"~~^^^; ~—^^|' L i A i y. D i l a vao each giai t t e n , t a r o the khong can phai dat an phi.i ma them bdt hang so dg tao dang phitdng t r i n h quen thuoc = D'^.
B a i t o a n 60. Gidi phiCdng trinh x^ + 4x2 (x - 2)2 D a p s6. X = -1 ± >/5.
B a i t o a n 6 1 . Gidi cdc phuong trinh sau :
- 12.
a + 25x2
(x + 5)2 25 _ 49 x2 (x - 7)'
= 11;
= 1;
6) x 2 + 9x2
[x - 3): = 14;
d) 9
4(x + 4 ) 2 + 1 = (2x + 5 ) 2 -
t o a n 62. Gidi he phudng trinh
( x + v/x2 + 1) {y + sji/ + 1) = 1 (1) 35
y + - + r r = o. - 4 ^ - - (2)
Giai. 'Oit-.vi kicn .T2- 1> 0 < ^ X > 1 ^ "^"^ "^^^'^ ^ ^^'""^
( x + \ / x 2 + 1) ( - X + ^ X 2 + l ) = I, \
(ij + v^y2 + 1 j + vy^TT) = 1.
K^t hdp cac he thvtc nay v6i (1), ta dudc | ^ + v / £ i ± i = "2/ + x/y^TI
y + v / y 2 + 1 = - X + ^ / ^ ^ ^ ^ .
Cong then ve liai dang thitc tren ta (hWc:
X + y + V^x^ + 1 + v / y 2 + 1 = -x - y + \ / x 2 + 1 + v / y 2 + 1 ;y = -x.
.T 35
Thay vao (2) ta ditdc x + ^ ^ ^ = — . ' (3)
• Neu X < - 1 I hi ve trai ci'ia (3) am. Vay x < - 1 khong thoa man (3).
• Neu .7; > 1 t h i bhih phifdng hai ve phUdng t r i n h (3) t a dUdc x^ +
Dat t =
2x •2 1225
\/,T2 - 1 x2 - 1 144 ..2
= 0 < ^ — — - + 2x2
> 0. Thay v;io ta dud(
1225 x 2 - 1 v/^2Tri 144 = 0.
f2 + 2t- 1225 _ 25
,..2 25
= — ^ 144x''-625x2+G25 ^
• 12
x2 = 25
^ V ; X > 1 va y =
en he i)hitdng t r i n h da cho co hai nghiem ( 3' ^3 j > ^
4' 4 If-.
^ a n g 2. Phu-dng t r i n h d a n g aj^ + n x + a + + a _ ^ x-^ + m x + a x"^ -\- px + a
5 5 - 3 x + - 5
X a:
5 5
x + - 4 x + - 6 X
P h u - d n g p h a p . D i f i u k i e n : {^2 ^ J ^ " ^ / , " _ f X e t x e m .x = 0 c6 p h a i \ n g h i e n i p h u d n g I r i n h k h o n g . TrUcfng h d p .r 7^ 0 :
^ , ; 1$.)../. , -, x + - + n x + - + 9 ( * ) ^ — f — + — i — =
.> ^ ' ' ' • x + - + m a - + - + p ' ^ ' X X
D a t y = .r + - , (lieu k i e n h/| > 2x/|ô|, t a c6 p h i t d n g t r i n h ^ " + _ ,
X • ,V + m ?y + 7>
G i a i p h U d n g I r i n h a n y san d o t i m x .
+ ' a i n • I I -T^^-Sx + S x ^ - 5 x + 5 1
B a i t o a n 6 3 . Gmi plinanq trmh = — . f i , x 2 - 4 x + 5 x 2 - 6 x + 5 4
G i a i . D i e u k i e n : x 7^ 1 va x 7^ 5. D e t h a y x = 0 k h S n g p h i i i l a n g h i e n i (!iia (1). X e t x^O. K h i d o
D a t y = X + ^ \y\ 2s/5. P h i t d n g t r i n h (2) t r d t h a n h :
y - 3 y-5 _ 1 ^ 2 ^ 1 y - 4 y ~6 4 _ jOy + 24 4
< ^ y 2- 1 0 y + 16 = 0 4^ f 2^ = 2 (1"^")
T i t do t a CO p h i t d n g t r i n h
' X + - = 8 4* x^ - 8 x + 5 = 0 ^ X = 4 ± ( t h o a d i e i i k i e n ) .
• '•if X
Vay p h i t d n g t r i n h (1) c6 t a p n g h i e n i : 5 = { 4 + y/n;4- s/u}.
Lvtu y. Cac d a n g p h i t d n g t r i n h san difdc giai m o t each t i t d n g t i t :
m x |_ "^-^ _ + " ' ^ + c px ^ ax2 + bx + d ax2 + cx + ci ^ ' ax^ + nx + c ax^ + ^ x + c ~
B a i t o a n 6 4 . Giai cac phuang trinh sau :
) + 3 x ^ 3 x 7 x
" M X2 - 8 x + 7 4x2 - l O x + 7 ^ x2 - 3 x + 1 ^ x2 + x + 1 ~ '
^ x2 - IQx + 15 ^ 4 x x2 + 5 x + 3 x2 + 4x + 3
''''' T? - 6 x + 15 ~ x2 - 12x + 5 ' ^ x2 - 7 x + 3 ^ x2 + 5x + 3 ~
S a n g t a c m o t s o d a n g p h i f d n g t r i n h p h a n t h i J c h i ? u t i . c a n h viec x a y d i f n g p h i t d n g t r i n h t i t he p h i t d n g t r i n h , viec x a y d i f n g hitdng t r i n l i t f t n h i t n g dftng t l i i t c d a i so co d i e u k i e n l a m o t t r o n g n h i J n g
^hi<''"S ph^-P S i u p t a t a o r a n h i t n g d a n g p h i t d n g t r i n l i h a y v a la. j i i i s . . ;
J. D u n g h a n g d S n g t h i J c ^ ^ ^
( a + b + c) 3 = + 6=^ + + 3 ( a + + c ) ( c + a ) .
Tft h a n g d a n g t h f t c n a y t h a y n g a y r a n g • , ^ , , : [ ; {a + h + cf = (t" + + (a + b){b + c){c + a) = 0 <^ a + b = 0
b+ c = 0 c + a = 0.
Van d u n g d i e u n a y t a de d a n g sang tac nen cac p h i t d n g t r i n h k h a t h i i v i . V i d u 1 . Chori a. b, c, chdng lian a = x - 2, 6 = 2 x - 4, c = 7 - 3 x . Sau do cho (a + b + c:)'^ = a'^ + + c"* ta duoc bai toan sau. , B a i t o a n 6 5 . Giai phuang trinh ( x - 2)^ + ( 2 x - 4)^ + (7 - 3 x ) ^ = 1. (1) G i a i . D a t a = X - 2, 6 = 2x - 4, c = 7 - 3x. T h a y vao (1) t a d u d c '
., ., '79 nôi
+b^ + c-^ = {a + b + c)\) Mat k h a c t a h i o n c6 d a n g t h i t c
{a + b + c)'* = a^ + b^ + c^ + 3{a + b){b + c)(c + a). (3) T i r ( 2 ) va (3) t a CO ( 0 + 6) ( 6 +c ) ( c +a ) = 0. D o d o
(1) ^ ( x - 2)^ + ( 2 x - 4f + {7- 3xf - [(x - 2) + ( 2 x - 4) + (7 - 3x)]^
^ ( 3 x - 6 ) ( 3 - x ) ( 5 - 2 x ) = 0 ^ 3 '=2-
•J, I
p h i t d n g t r i n h (1) co t a p n g h i e n i : 5 = | 2 ; 3 ; ^ | .
LiTu y, Vc3i b a i t o a n t r e n , each t i t n h i e n n h a t co le l a k h a i t r i e n r o i t h u ve Phitdng t r i n h bac b a . S a u day t a se t h i e t ke m o t b a i t o a n m a viec k h a i t r i c n
gap vo v a n k h o k l u l i i . , .
^ ằ d u 2 . Chon a = x2 - 4x + 1, 6 = 8x - x^ + 4, c = x - 5. Sau do cho (a + 6 + (.)•' = + + ta dupc bai toan sau.
t i v . ' ' h iirifra ^nfv-- t o a n 6 6 . Giai phuong trinh
(x^ - 4x + 1)3 + (8x - x^ + 4f + ( x - 5f = 125x3.
2. D u n g m e n h de
- + ^ + - = <^ ( a + + c ) ( c + a ) = 0.
a o c a + o + c
C h i i n g m i n h . Ta co ; i t . . (a + c ) ( c + a ) = 0 <^ (a + ; ; ) ( 6 c + a c + c ; ' ' ^ ) = 0
M a t k h a c '_>'^
G i i V •Sj.hbil 1 1 1 1 a6 + ; ; c + m 1 , - + 7 + - = ^ . ^ ^ j = , • , 4- 4
o. D c a + h + r abc a + b + c •
^{a + b + c){ab + be + ca) = abc
ô-a^6 + o^c + abc + b^a + b^c + ft6c + + c^o + a6c = abc
k>,rnr*> ^2abc+a'^b+a'^c + b'^c + b^a + c'^a + c'^b = 0. (2) TCt (1) va (2) SUV l a iiienli de t i e n diiiig.
V i d u 3. Tii: •incnh dc trCn, lay a = x - 8, b = 2.r + 7, r = 5.T + 8, r/vwc bai toan sau.
B a i t o a n 67. Giai pkildng trinh
1 1 1 1
+ W—^ + T-T^ = Z-^- (1)
X -- 8 2x + 7 5x + 8 8x + 7 7 8 7
G i a i . D i e u k i c n : x ^ 8,x^ ~ o ' ^ " O - '^^ chitog m i n h ditdc 2 5 8
1 x = (?
X = : - y . 15 Do (2) nen t a c6
(1) o (x - 8 + 2x + 7)(x - 8 + 5x + 8)(2x + 7 + 5 x 4 - 8 ) = 0 < ^
PhiTdng t r i n h c6 tap nghiem : S = |^'^'
L i f u y. Doi vdi bai toan, neu khong biet silt dung menh de (2) t h i se rat kho khan do t h n l a Irii giai.
3. D u n g m e n h de :
" N e u x i / : = 1 v a x + y + ^ = i + - + - t h i (x - l ) ( y - l ) ( z - 1) = 0" •
X y z
C h i f t i g m i n h . Tit gia thiet t a c6 xyz — 1 vh x+ y + z = xy + yz + zx. Do do (x - l){y - l)iz - 1) = (x - l){yz - y - z + l) i v ^ w a h . •
= x y z - xy - xz + x - yz + y + z - 1 = x + y + z - {xy + yz + zx) = 0.
V i d u 4. Chpn a, b, c sao cho abc = 1, chdng han a = 2 x - l , 6 = 5 x - 3 , c = 1
10x2 _ i i a ; + 3' rfnrr • Kht do
u + b + c = 7x-A + 1
10x2 - l l x + 3 1 1 1 _
a ^ 6 ^ c ~ 2 x - l • 5 x - 3 ^ + . „ + 10x2 - l l x + 3.
I l l
Do do a + b + c = - + - + - t'Udng duong vdi - I ' l a b c
^ + — ^ - — + 1 0 , T 2 - 1 1 X + 3 = 7 X - 4 + ^ 2x - 1 5x - 3
Va?/ to CO 6az <odn sow.
B a i t o a n 68. Gidi phudng trinh
1 1 1
1 0 . T 2 - l l x + 3 10x2 - l l x + 3"
1 0 x 2 - l l x + 3 2 x - l 5 x - 3 +
G i a i . Dieu fc^n: xi^ - ,x^ - . Dat 3 1
o 2
+ lOx^ - 18x + 7. (1)
a = 2x - 1,6 = 5x - 3, f = 1
10x2 _ i i a , + 3' f
Khi do a6c = 1 va tir (1) suy ra a + 6 + c = i + 7 + - . Ta chiing m i n h diTdc
^ ' a b c ket qua : Neu xyz = 1 va X + y + 2 = - + - + - t h i (x - l){y - l){z - 1) = 0.
x y z Tit do
( l ) ^ ( 2 x - l - l ) ( 5 x - 3 - l ) ( ^ ^ ^ , _ \ ^ ^ . ^ 3 - l ) ^ 0 r x = i
X = - : ' . 10.7;2 - l l x + 2 = 0
X = 1 4 1 1 ±
Phirong t r i n h (1) c6 tap nghieni S = y-'t' ^20^
4. D u n g m e n h d § : V d i a, 6, c l a c a c so thulc t h o a m a n a + b + c. = a t h i vj>jj rxiihh. I'i.
1 j _ _1_
C h i J n g m i n h . Vc3i a + 6 + c = 0 ta c6 1 1 1
- + - + -a 0 c 1 1 1 n
1 1 1 - + J + - a b c
( 1 1 \
^ + 7 - + —
a?- l)^ \ab be ca J
1 1 1 2{a + b+c) 1 1 1
= _ -I I I '- — \ .
a'^ 6^ c2 abc d?- 6^ c?' V a y i / ^ + j ^ + ^ =
1 1 1 - + - + - a b c
B a i t o a n 69. Gidi phuong trinh 1 +
1
(1).
G i a i . Dieu kien ^ ^ ~ ^ } ' chi'riig m i n h ditdc : V 6 i a, b, c la cac so thi.te thoa man a + b + r = 0 t h i \/ -2 + ^ + -2 = 1 1 1
- + - + -a b c . Ta CO
( 1 ) ^ 1
+ 1 + 1
{2x-iy {-3x-iy {x + 2y (x + 2) V i (2x - 1) + (-3x- - 1) + (x + 2) = 0 nen
' 1 2 • (2)
1 + 1 7 0 + 1
y ( 2 x - 1)"^ • ( - 3 x - 1)'^ {x + 2f Vay (2) tUdng ditdng vdi
+ 1 1 +
(2a,--1) ( - 3 x - l ) {x + 2)
1 + 1 + 1 \x + 2\
( 2 x - l ) ( - 3 x - l ) (x + 2)
( - 3 x - l ) ( x + 2) + (2x - l ) ( x + 2) + (2x - l ) ( - 3 x - 1)
( 2 x - l ) ( - 3 x - l ) ( x + 2) |x + 2|
-7x^ - 3x - 3
( 2 x - l ) ( - 3 x - 1) = 2 <^ x^ - X - 1 = 0 < ^ X =
1 ± \/5
Vay phirdng t r i n h c6 nghieni x = 1 ± \/5
]>Jhan x^t- Qua cac vf du tren, t a c6 the hinh dung co ban viec svt dung
aang thiic de xay d^Ing phUdng t r i n h . H y vong dua vao von hieu biet va kha iiaiig sang tao cua minh, ban doc co the tao ra nhfnig phudng t r i n h dep mat va doc dao hdn nfra.
>lf3 rill,fl!
Chtfdng 3 ; ; /
Phijfdng trinh, bat phLfdng trinh chiJa can thiJc
3.1 Phi^dng trinh ± ^/B{^ = ^/C {x).
(trong do A[;x), B{x), C{x) la car da tln'lc co bar khong qua 3).
P h t f d n g p h a p . Lap phUdng hai ve, sut dung cac hang d i n g thiic sau : {a + bf = a^ + b^ + 3ab{a + b).
{a-b)^ = a^ ~3ah+3ab'^-b^ = a^-b^-3abia-b).
B a i t o a n 1 ( D e t h i H S G cap t i n h G i a L a i - 2 0 1 0 ) . Giai phUdng trinh
^ 1 5 x - l + v^l3x + l = 4 ^ . (1) G i a i . Lap phudng hai ve phildng trinh (1) ta dudc
15.T - 1 + 13.r + 1 + 3 iJ/(15.T - l)(13x + 1) {^Ihx - 1 + V'13.T + 1) = 64.T.
M a ^15x - 1 + s/13x + 1 = 4^/5 nen
15x - l + 13a; + l + 3v/(15x - l ) ( 1 3 x + l ) . 4 ^ = 64x
^ 1 2 ^ x ( 1 5 x - l)(13x + l ) = 36x ^ ^ x ( 1 5 x - l)(13x + 1) = 3x
<=^x(15x - l)(13x + 1) = 27x3 ^ J X =^0_ ^^^^^^ ^ ^ ^ ^ ^^^^
r - r — n f
L T 6 8 x ° + 2 x - l = 0
Thi'r lai thay x = 0, x = - j ^ , x = la tat ca cac nghiera ciia (1).
L i m y. Phep the ^15x - 1 + i y i 3 x + 1 = 4 ^ c6 the dan tdi mot phiWng trinh he qna, do do sail khi t i m difdc nghiem ta phai thay vao phitdng trin'' da cho d§ kieni xem c6 thoa man hay khong.
p a l t o a n 2. Giai cac phuong trinh sau , j , . . ^
a) ^^2^^ + v ' ' ^ ^ = v ^ S ^ I ; 6) sJ^FTT + v ^ ^ ^ = v ^ .
Dieu kien x € R. Lap phUdng hai ve phUdng trinh da cho ta ditdc
2 x - l + x - l + 3 v / 2 r ^ v / ^ ^ ( v ^ 2 7 ^ + S / ^ ^ = 3x + 1
<f4'S /2 ^^S /^ir^(v /2^^+ S / ^ ^ = 1.
ma ^ 2 x - 1 + ^ / ^ ^ = ^ 3 x + 1 nen ta c6
s!/2^^s/^^^3/3lTT = 1 'ytn'-
^ (2x - 1) (x - 1) (3x + 1) = 1 <^ X e |o; ^1 .
ThiJt lai thay x = ^ la nghiem duy nhat ciia phUdng trinh da cho.
6) Dap so : Tat ca cac nghiem la 0, \/5
2 ' 2 • f i ; ; - ,
Bai t o a n 3. Giai phuang trinh v^2 + x + x2 + ^/2~x-x^ =
D a p so. X = l , x = - 2 . "
Bai t o a n 4 (PhvTdng p h a p h a m l i e n t u c ) . Giai bat phuang trinh
s^^^rn + s/e^iTTT > ^ ^ 2 7 ^ . (i)
G i a i . Tap xac dinh R. Ta giai (1) bang each xet dau ham so lien tuc tren
Taco
<=ằ8x + 2 + 3 v/ 2 x T T v/ 6^ T T ( v ' ^ ^ T T + v/e^TT) = 2x - l . Thay the s/2xTl + v^6x + 1 bdi v^2x - 1 vao (3), ta dUdc
(2) (3)
6x + 3 + 3 v^2x + 1 v^6x + 1 ^ 2 x - 1 = 0
^2x + 1 + s>^^nn;v/6^Mn:^y2^r^=o
<!^s/2^Ti (^(2x + i)2 - ^3/feTTv/r=^) = 0 - M
s /2xTT=_o X = - -1
^ ( 2 x + i)2 = s y S J T T ^ T ^ ^ [ jg^2 i o
1 x = - - -
X = 0.
T h a y x = 0, x = - ^ vao (2) thay 3: = - ^ la nghiem d u y nhat ciia ( 2 ) . H a m so f{x) lien tuc va phifdng t r i n h 2 f{x) = 0 CO nghiem d u y n h a t tren R. 1 i
tren R. L a i c6 / ( O ) = 3 > 0, / ( - I ) = ^ + - s/=3 < 0. V?ly / ( x ) >
Vay /(.T) CO k h o n g qua m o t Ian doi d a u ./(-T) <*
tren R. L a i c6 / ( O ) = 3 > 0, / ( - I ) = s / ^ + - 0 <^ X- > T a p nghiem cua (1) la + 0 0 ^. 4'
3.2 Phu-dng trinh [ax + 6)" = p \ / a ' x + b' + qx + r.
vdi .7: la an so ; p, q, r, a, h, a', b' la cac hang so ; paa' ^ 0 ; n e {2,3}.
3.2.1 Phvfdng phap giai.
• D a t an p h u :
- D a t \/a'x + b' = ay + b neu pa' > 0.
- D a t s/a'x + b' = -{ay + b) neu pa' < 0.
• B a i toan d a n den giai he phurtng t r i n h h a i an (thUrJng l a he d o i xi'mg loai 2) d o i v d i x va y. De giai he nay, t a t r i i h a i phudng t r i n h (hoac cong hai phirdng t r i n h ) va do y don t i n h ddn diou ciia h a m so.
• T h u a t d a t §,n p h u n h u t r e n goi la t h u a t dat "an phu doi xutig".
C h u y 1. Phicanfj trmh dang (ux + ^ ) " = p \/a'x + b'+ qx + r rd tM giai duclr.
phudng phdp nay thiidng thoa man dieu kien :
• Neu b = 0 ihir = b'.
( a' -q b'-r , , , , p = = —;— (neu pa' > 0)
• Neu b^O thi II a' - q b' - r , . , „^ 0, b I -p = = —;— [neu pa < 0).
1-./ , „ > a b
3.2.2 M o t s6 b a i toan r e n luyen v a nang cao.
B a i toan 5 . Giai phuang trinh \/2x + 15 = 32x2 ^ 323. _ 20.
G i a i . D i e u kien 2x + 15 > 0 <^ x > - 7 , 5 . Phifdng t r i n h d a cho v i e t l a i
v'2x + 15 = 2(4x + 2)2 - 2 8 . (1) Dat s/2x + 15 = 42/ + 2, v(3i dieu kien 4j/ + 2 > 0 <^ y > - 0 , 5. K h i do
+ 2f = 2x + 15. K e t hdp vdi (1) t a c6 he
/ (4i/ + 2)2 = 2 x + 1 5 (2)
1 (4x + 2)2 = 2 t / + 15. (3) rQfl thco t i n i g vc cua (2) va (3) t a dUOc
(4t/ + 4x + 4)(42/ - 4 x ) = 2 ( x - y) ^ {x - y)[l + 8 ( x + j / + 1)] = 0.
, K h i X = y, t h a y vao (3) dudc (4x + 2f = 2x + 15 ^ IGx^ + 14x - 11 = 0.
Giai phUdng t r i n h nay dUdc h a i nghiem la x = - va x = . S o sanh v d i 1 , ^ dieu kien cua x va y c h i c6 nghiem x = - thoa m a n .
• K h i 1 + 8(x + y + 1) = 0 y = - X - - , thay vao (3) t a dUdc 8
(4x + 2)2 = - 2 x - ^ + 15 <^ 64x2 _^ 72X - 35 = 0.
9 i -^^221
Giai phu'dng t r i n h nay dUdc hai nghiem la x = — . So sanh vdi dieu Q _l_ ^ 2 2 2 16
ki^n ciia x va y c h i c6 nghiem x = — thoa m a n . Vay phirdng t r i n h l b
. . . , , . , 1 . - 9 + v/22T
da cho CO h a i nghiem l a x = - va x = . 2 16
Li/u y. Co t h g tiep can b a i toan nay thong qua cong cu dao h a m n h u sau : Xet ham so / ( x ) = 32x2 ^ 2>2x - 20. K h i do / ' ( x ) = 64x + 32 = 32(2x + 1).
V%y, dat V2x + 15 = a ( 2 y + 1), t a c6 h?
2x + 15 = a2( 2 y + 1) 2 ^ / + = / ^ ^ ^ + 28 a ( 2 y + l ) = 8 ( 2 x + l) 2- 2 8 ^ 1 2y + 1 = - ( 2 x + 1)2 - — .
a a
Ttt he nay, de thay rang c i n chon a sao cho a2 = — <^ a = 2.
a
(1) B&i toan 6. Giai phudng trinh 4x2 ^. ^'3^. + j -f- 5 = I 3 x .
GiSi. D i e i i kien x > -~. phirdng t r i n h da cho v i e t l a i
o
(2x - 3)2 = - V / S ^ M H : + X + 4.
^ ^ t v'3x + 1 = - ( 2 y - 3 ) , dieu kien - ( 2 y - 3) > 0 y < 1. 5. K h i do
^ + 1 = (2y - 3)2. K e t hdp vdi (1) t a c6 he
( 2 x- 3 ) 2 = 2y + x + 1 (2) • ^ ' ( 2 y- 3 ) 2 = 3x + l (3)
Tiu; theo ve cac phudiig t i i n h (2) va (3) t a dirdc
2(2x + 2y- 6){x - y) = 2y -2x^{x- y){2x + 2y - 5) = 0.
• Tritdng hdp y = x. Thay vao (2) t a durtc ••. - -
4a;2 - 12a; + 9 = 3x + 1 <^ - 15x + 8 = 0 <^ a; = — . 8
Ket hdp vdi digu kien cua x vh y ta chi nhan nghiem x = — — . 8
• TrUdng hdp 2x + 2y - 5 = 0 <^ 2y = 5 - 2a;. Thay vao (3) ta ditdc
: (2 - 2xf = 3.7; + 1 4.T2 - ll.-r + 3 = 0 <^ x = l i A ^ ^ . ^
K c t hdp vcfi dicu kicn ciia x va y ta chi nhan nghiem x = — \
. I . ^ u 1^ n + V73 1 5 - ^ nghiem cua phUdng t r m h la x = , x — .
8 8 B a i t o a n 7 ( D e nghi O L Y M P I C 3 0 / 0 4 / 2 0 0 6 ) . Giai phuctng trmh
Sy6.r + l = 8: r^ - 4. T - l .
G i a i . Tap xac diuh cua phudng t i i u h la E . Dat ^Qx + 1 = 2y. Ta c6 he 8x^ -4x-l=2y ^ / 8x^ = 4x + 2y + 1 (1)
6a; + l = 8y^ \82/'^ = 6 x+ l . (2) Lay (1) trir (2) theo ve t a dUdc
8{x^ - 7 / ) = 2(y - x) ^ (a; - y)[4(x2 + xy + y^) + I] = 0 ^ y = x.
Thay y = x vao (2) t a du'dc
8x^ - 61 = 1 <^ 4x^ - 3x = cos ^ .
SiJt dung cong thiJc cos a = 4 cos^ — - 3 cos - , ta c6 o 3 cos x = 4 cos — - 3 cos - ,
O u7 i>
T T T ^ 77r 77r
cos — = 4 cos — — 3 cos -—,
O i7 L7
cos = 4 cos — — 3 cos — .
O tj if
= (JUS 1 •'' — ^ 1 ^ = cos la t a t ca cac nghiem ciia phUdng t r i n h TT 7n
V W H V a
jo) va cung la t a t ca cac nghiem cua phUdng t r i n h da oho. ô T ufu y- Phep dat \/6x + 1 = 2?/ da ditdc noi d phan phUdng phap giai. Tuy
hien ncu qucn phudng phap t h i t a van t i m ra dirdc phcp dat nay di.ra tren guy luan t u nhien n h u sau : Gia sijf s/6x + 1 = ay + b. K h i do
ay + 6 = 8x^ - 4a; - 1
6x + 1 = a^y^ + Sa^by^ + dab'^y + b^
r^ay -6x + b - l ^ 8x^ - a^y^ - ^a^b-y"^ - 4x - 3ab^y - I - b^
=> (8x^ - a^y^) - 3a^y^ + [2x - (a + 3ab'^) y] - {b^ + 6) = 0.
pg suy l a dUdc x - y = 0, ta can c6
r a + 3ab'^
I 3a
•'J I ' ! . I'll >
a-^ a + Sab-" ( b = 0 ( , „ i ' ' "
Vay t a c6 phep dat s/6x + 1 = 2?/. Ta con c6 the giai each khac nhif sau : PhiTOng t r i n h viet lai
6x + 1 + v^6x+ 1 = (2x)^ + 2x. (3) Xet ham so J{t) = + tyt e R. V i f'{t) = 3/^ + 1 > 0,Vt g R nen ham so fit) dong bien tren E . M a phUdng t r i n h (2) viet lai / {^6x + l ) = /(2x) nen u6 tUdng dUdng \/6x + 1 = 2x <^ 8x^ - 6x = 1 4^ 4x^ - 3x =
Bai toan 8 ( C h o n doi t u y e n T p H o C h i M i n h duf t h i quoc gia n a m hoc 2 0 0 2 - 2 0 0 3 ) . Giai phUdng trinh ^ 3 x - 5 = 8x^ - SGx^ + 53x - 25.
G i a i . Tap xac d i n h R. PhUdng t r i n h viet lai
= (2x - 3)^ - X + 2. (1)
D^t 2y - 3 = ^/Zl^. K e t hdp vdi (1) t a c6 h§
( 2 y - 3 ) 3 = 3 x - 5 (2) ( 2 x - 3 ) 3 = x + 2 y - 5 (3)
{
Lay (3) trir (2) theo v 6 t a diMc
2 (x - y) \{2x - 3)2 + (2x - 3) {2y - 3) + (2y - Zf] = 2{y - x)
X - y = 0 (4)
(2x - 3)2 + (2x - 3) (2y - 3) + (2y - 3)^ + 1 = 0. (5)
i Ta CO (4) ^ y = x. Thay vao (2) t a dUdc
• (2x -3f = 3x -5 ^ 8x^ - 36x^ + 54x - 27 = 3x - 5
<::>(x - 2)(8x2 - 20x + 11) = 0 ^
3B^
X = 2 .
5±V3
4 rnn ( M t t f i
'Do A'^ + AB+ 3"^ = {A+--- + > 0 nen (5) khong the xay ra.
5 i PhiTOng t r i n h c6 ba nghiem x = 2, x = —-—.
Lvfu y. T h e m mot phuang phap nfra dc t u n ra phcp dat 2j/ - 3 = v^3x do la : Ta dat ay + b — \/3x - 5, vdi a, b se t i m sau, sao cho cung vdi p h u d n g tririh da cho t a t h u diTdc mot he doi xiing loai I I , hoac la mot he gan doi xiing loai I I (trijf hoac cong hai phudng t r i n h t a t h u dvtdc x = y ) . Ta c6
Vay ta chon avkb sao cho | 2 ~ ^ ^ { a = 2 ^ T i t do phcp doi bicii [ - 3 - 6 = 0
la 2y - 3 = v^3x - 5. Ciing c6 the tiep can phep doi bien 2y - 3 = v^Sx - 5 thong qua cong cu dao ham nhit sau : X c t ham so
/ ( x ) = 8x^ - 36x2 ^ _ K h i do
/'(x) = 24x2 - 72x /"(x) = 48x - 72 = 24 (2x - 3 ) . Vay dat ^ 3 x - 5 = a(2y - 3). Ta t h u dUdc he
/ 3x - 5 = a3(2y - 3)3 ^ f
\y - 3) = (2x - 3)3 - X + 2 ^ \
3x - 5 = a3(2y - 3)^
(2x - 3)^ - X + 2 + 2 ^ 1 2 y - 3 = i
a
Tvc he nay, de thay r^ng can chon a sao cho = — =^ a = ±1.
3.2.3 Phifdng phdp sang tac bai toan mdi.
V i d u 1. Ta se sang tdc phicang trinh c6 it nhat mot nghiem theo y muSn- Xet X = 3. Khi do 2x - b = 1 => (2x - 5)^ = 1 ' ^ ^ ^ x - 2. Ta mong muon CO mgt phuang trinh chtia {ax + b)^ va ch-da \/cx + d, hon nUa phuong trinh
,fiay diicfc gidi bang each diCa ve he "gan" doi xiing loai hai {nghia la khi tril, cQTig hai phuang trinh, cua he ta c6 thUa so [x - y)). Vay ta xet he •> .
r ( 2 y - 5 ) 3 = x - 2 (i)
• \x - 5)3 = - x + 2y - 2. (ii) ,.. , ^xO'S • ,>
jVew CO phep dat 2y - 5 = \/x - 2, thi sau khi thay vao phUdng trinh [ii) dupe gj;3 - 60x2 igQ^ _ = - X + ^ x ~2 + 5 - 2 . Ta c6 bai todn sau.
g a i t o a n 9. Gidi phuang trinh ^x-2 = Sx^ - eOx^ + 151x - 128.
G i a i .
C a c h 1. PhUdng t r i n h da cho dUdc viet lai ^ x - 2 = ( 2 x - 5)^ + x - 3. (1) Dat 2?y - 5 = v^x - 2. Ket hdp vdi (1), ta c6 he
( 2 y - 5 ) 3 = x - 2 (2) (2x - 5 ) 3 = - X + 2y - 2 (3)
Lay (3) trir (2) theo ve ta dUdc - 2 (x - y) [(2x - 5)2 + (2x - 5) (2y - 5) + (2y - 5)2] = 2(y - x)
r x- y = 0 (4)
[ (2x - 5)2 + (2x - 5) (2y - 5) + (2y - 5)2 + 1 = 0. (5)
• Ta CO (4) y = X . Thay vao (2), t a dudc
(2x - 5)3 = X - 2 <t=> 8x3 _ gQ^2 ^ j 4 9 ^ - 123 = 0 "
(x - 3)(8x2 - 36x + 41) = 0 X = 3.
•DoA^ + AB + B'^ = (
- I + 302 > 0 nen (5) khong the xay ra. 11? si / Phudng t r i n h c6 nghiem duy nhat x = 3.
^ 0 phuang trinh cd nghiem duy nhat x = 3 nen ta nghi den phuang phdp svt dung tinh dOn dieu cua ham so nhu sau :
Cach 2. Tap xac dinh E. Dat y = s / J ^ . Ta c6 h f / 8x3 - 60x2 + i51x - 128 = y
\; = y3 + 2
*^Oiig ve theo ve hai phitdng t r i n h cua he, t a ditdc
8x3 _ gQ^2 ^ j 5 2 x - 128 = y3 + y + 2 ' '
<^8x3 - 60x2 ^ j5o^ - 125 + 2x - 5 = y3 + y
<i=>(2x - 5)3 + (2x - 5) = y3 + y. (*)
Xet ham s6 f{t) = e + t. Vi f'{t) =3f + l>Q,Vt&R nen ham / dong biln tren E. Do do (*) viet lai f{2x - 5) = f{y) ^ 2 x - 5 = y. Bdi vay
{2x - 5) = ^yr^ <^ (2x - 5)3 = X - 2
<^8x^ - 60x2 _^ j49^. _ 123 = 0 (x - 3 )(8x2 - 36x + 41) = 0 <^ x = 3.
Phildng trinh c6 nghi§m duy nhat x = 3. ^ .... ^ _
Vi d u 2. Xei mpi phuong trinh bac ha nao do, chdng han xet Ax^ + 3x = 2.
Phitang tnnh nay tudng duang
Sx^ + 6x = 4 Sx^ = 4 - 6x <^ 2x = v'4 - 6x.
Ta "long ghep " phUdng trinh cuoi vao mot ham dan dieu : . j
(2x3) ^ 2x = ^ifl^^ + 4 - 6x <^ 8x3 + 8x - 4 = ,^435^. •
Ta dUdc bai toan sau.
Bai toan 10. Giai phUdng trinh 8x3 + 8x - 4 = ^4 _ 6x.
Giai. Tap xac dinh cua phUdng trinh la K.