1.11.1 P h i f d n g p h a p difa v e phifdng t r i n h t i c h , phifdng p h a p d a n g c a p h o a d\ia v e phu'otng t r i n h t i c h .
Doi vcfi phudng phap nay t h i viec sang tac phudng trinh la rat de dang, t a chi can xet f{x).g{x) = 0, sau do t a nhan vao va bien ddi de khong Con tich iiiia. Con vice giai thutJng la di ngUdc lai qua trinh sang tac, phan tich thanh PhiTdng trinh tich. Tuy nhien khi sang tac phifdng trinh t a ciing khong nen qua tuy tien, m a nen xay dvtng nen nhiing phudng trinh dufa ve tich c6 dac thvi rieng, hoac dai dien cho mot phutdng phap phan tich thanh tich nao do.
B a i toan 149. Giai phKdng trinh 2 v T ^ - ^/TTx + 3v/l - = 3 -
G i a i . D i e u k i e u - 1 < a; < 1. P h U d n g t r i n h v i e t l a i
(1 + x) + 2 (1 - x ) - 2Vl - X + v/1 + X - 3 \ / l - x-2 = 0. (1) Dat V s/r+x > 0, v = ^/l - X > 0. Thay vao (1) t a d u d c
+ 2v^ - 2v + u-3uv = 0'iF^{u-2v)(u-v + l) = 0<^
• K h i u — 2v, t a C O
u = 2t;
U = V — 1.
" ' ' \ / l + X = 2y/l - x < ^ l + x = 4 - 4 x < ằ 5 x = 3 ô > x = - . 5
• K h i = u + 1, t a C O
\ / l + 3 ; + l = \ / l - x < ^ 2 \ / l + x + 2 + x = l - x •
. ^ 2 v T T ^ = - 2 x - l ^ { ^ x = - ^ .
{- ,3 \/3 5 2 - —
L i m y. Phop phan tich (1) dime t i m ra n h u sau : Chon a, (i sac cho
- x + 3 = Q ( \ / r ^ +/3(vTT^)
^ - x + 3 = ( - a + ^ ) x + a + /3=^{ ^ { ? = 1.
Con phcp bicu doi
+ 2(;^ - 2v + u- "iuv = 0 <^ ( u - 2i;) ( u - y + 1) = 0
dUdc t i m ra bang each t a xeni + 2v^ ~2v + u - Zuv = 0 la mot phudng t r i n h bac hai theo u (con v la thani so) c6 A - (u + 1)^, t i t do t i n h diTdc u = 2v, u = V - \.
V i d u 1. Bung each him doi (\/xT8 - 3 x ) [s/x + 8 - ( x + 2)] = 0, ta dugc bai todn sau.
B a i t o a n 150. Gidi phuang trinh
3x2 + 73; + 8
G i a i . Dieu Icien x > - 8 va x 7-^ Ta c6
( l ) < ^ x + 8 - ( 4 x + 2)^JT8 + 3x2 + 6x = 0. (2)
^^^^
"i.-'i- "f:: 'J
at t = y / x T S . T h a y vao (2), t a difdc
f2 - (4x + 2) / + 3x'^ + 6x = 0.
y^^t bie t thirc
A ' = (2x + 1)^ - (3x2 + 6x) = x2 - 2 x + 1 = ( x - 1)^ V/A' = | X - 1| . Vay t i l (3) t a ditdc t = 3x hoac i = x + 2. N g h i a l a
(3)
v/x + 8 = 3 x ^ v / : i ^ = x + 2
3x > 0
9x2 - X - 8 = 0
X > - 2
x2 + 3x - 4 = 0
x = I.
Yay X = 1 l a n g h i e m duy n h a t ciia p h U d n g t r i n h da cho.
Lvtu y. Se n h a n h h d n n e u b i e n d o i (2) t h a n h p h i t d n g t r i n h t i c h :,
(2) ^ {V^i^ - 3x) [V^TfS - (x + 2)] = 0. ,,, . ,
V i d u 2. Xet mot phuang trinh bac ha c6 nghiem "dep " ; - 3 i + 2 = 0. Tit day chon t = - ta duac x^ - 'ixy^ + 2y^ = 0. Lai chgn y = \/x + 2, ta duac
- 3 x ( x + 2) + 2yJ{x + 2f = 0 ^ x^ - 3x'^ + 2y/{x + 2f - 6 x = 0.
Ta CO bai todn sau.
B a i toan 151 ( D e nghi O l y m p i c 3 0 / 0 4 / 2 0 1 1 ) . Gidi phuang trinh x^ - 3x2 ^ 2y/ix + 2f - 6x = 0.
G i a i . Dieu kien x > - 2 . Phifdng trinh viet lai
x^ - 3 x ( x + 2) + 2^{x + 2f = 0. (1) Dat y = v / x T 2 , dieu kien ?/ > 0. Thay vao (1) ta dit0c
x^ - 3x2/2 + 2 i / = 0. (2) Neu 2/ = 0 t h i t i f (2) c6 x = 0, mau thuSn vdi y = v / x T 2 . Xet y 7^ 0. K h i do
'fj- 3
( l ) ô ^ ^ - 3 . - + 2 = 0 ^ - = - 2 h o a c - = l . V y y
• K h i - = 1, t a C O y = x , do do y
K h i — = —2, t a c6 2y = -x, do do y
Phttdng t r i n h c6 tap nghiem la 5 = { 2 , 2 - 2\/3}.
L i f t i y. Ta c6 the t r i n h bay ng^n gon hdn n h u sau :
- 3xy'^ + 2y3 = 0 O (x - yf{x + 22/) = 0 <^ X = y
X = '-2y.
1)
B a i t o a n 152. T-H: phudng trinh bdc hai - 27t + 50 = 0, chon t = -, duac
u
- 27vu + 50u2 = 0. Tiep tuc chon u = 3'^'+'=, v = 3^^"^ ta diCcfc
g^ 2 x- i _ 27 3 x 2+ 5 x- 2 ^ 50.9^'+^ = 0 ^ 81^^-^ - 3^'+^^+^ + 50.9^'+'= = 0.
Ta duac bai toan sau.
B a i t o a n 153. Gidi phMng trinh S^'+^^+i - 50.9^'+^ - Sl^^-^ = 0.
G i a i . Phirong t r i n h v i l t lai
3 i ( 2 x 2 + 2 x) + i ( 8 x- 4) + 3 _ 50.32x2+2x _ 3 8 X - 4 ^ Q
Dat u = 3^'+^, V = 3''^-2 K h i do (1) t r d thanh
27uv - bOu^ (25u -v){v-2u) = 0^
Do do
(1)
25u = V
V = 2u.
25 3x2+x ^ 3 4 x - 2 34^-2 =: 2.3^'+^
x 2 - 3 x + 2 + log3 25 = 0 x2 - 3x + 2 + log3 2 = 0 ^
+ X + log3 25 = 4x - 2 x2 + X + logg 2 = 4x - 2
x2 - 3x + logg 225 = 0 (2) x2 - 3x + log3 18 = 0. (3) Do (2) va (3) v6 nghiem nen phitdng t r i n h da cho vo nghiem.
L i f u y. Phep phan tich (1) diMc t i m ra nhrt sau : Ta can t i m a, /3, 7 sao cho x^ + 5x + 1 = a (2x2 ^ 2x) + p (8x - 4) + 7
r 2a = 1 / I 1 \
=^ 2a + 80 = 5 , ^ ( a; / 3 ; 7 ) = - ; - ; 3 .
1 - 4 / 3 + 7 = 1 \ 2 y
V i d u 3. Xet x = 2 va y= I, khi do
20y'^ (x^ - 2/2) = Sx^ (x^ +1/^) ((fing cap bac A)
Thay x = 2 vd y = 1 vdo ( * ) to diCcfc k = 1, vdy c6 bai toan sau.
t o a n 154 ( C h o n doi t u y e n T H P T c h u y e n L a m S d n , T h a n h H o a , nam hoc 2 0 1 0 - 2 0 1 1 ) . Gidi he phuang trinh , ^,
r 2 y( x 2 - y 2 ) = 3x (1) " ' ^ (
• \(x2+2/2) = iOy. (2) . t I
G i a i . Ta thay neu x = 0 t h i 1/ = 0 va ngUdc lai nen he plutdng t r i n h da cho CO nghiem (x,2/) = (0,0). Xet trudng hdp x / 0 va 2/ 7^ 0. Chia ve theo ve phitdng t r i n h (1) cho phudng (2), t a dudc
2 y( x 2 - 2 / 2 ) _ 3x d t k O •4"'- ^
X (x2 4- 2/2) IO2/
4^20y2 (x2 - 2/2) 3^.2 (^.2 ^ y2^ ^ 3^.4 „ 17^.2^2 _^ 202/4 = Q
<^ (x^ - 42/2) (3x2 _ 5^2^ = 0 <^ = 42/2 hoac x2 = -y^.
o
Neu x2 = 42/2, he da cho t r d thanh
2y.3f = 3x r 2y3 = X ^ f 22/^ = x ^ / x = ± 2 x.52/^ = lOy ^ I X2/ = 2 ^ \4 = 2 ^ \ - ± 1 . , . Neu x2 = -2/2, he da trcl thanh
0
4^2 = 102/ ^4.x2y =
. 47/3 = 9x .
15 ^ 1 162/4 = 135 ^ ^
X = ±
2/ = ± 15 2v/l35
^yi35
Vay he c6 5 nghiem la (do tu: 2y^ = x, 4y^ = 9x suy ra x va 2/ ciing dau) (x,2/) = (0,0), ( 2 , 1 ) , ( - 2 , - 1 ) , 15 15 v/l35
2 1 ^ / 1 3 5 ' ^ y ' \ 2 ^ ' ' 2 y
1.11.2 S i J d u n g k h a i t r i l n N h i thiJc N i u - t d n de s a n g t a c m o t s 5 h e p h i f d n g t r i n h k h o n g m a u mu'c. j ; V i d u 4. Xet
ix + y = ^ {x + yf = 5 ^ f 5 = x^ + 3x2?/+ 3x2/2 + (1)
\ x - l = Y ^[{x-yf = l ^ \ = x 3- 3 x 2 j / + 3 x y 2- y 3 (2) Cong (1) vd (2), lay (1) trii (2), i/ico w to dUdc
{ 5 = = x3 + 3xy2 ^ ^ = .x2 + 3?y2 (3) 3x2y + y3 S 2 ^ 3 ^ 2 + 2 (4)
I y
Lay (4) trii ( 3 ) , lay (3) cong (4) ta duac - - - = 2 x 2 - 2 y 2
y, ^ ^ - + - = 2(2^2 + 22/2)
Ta duac hdi loan sau.
B a i t o a n 155. Gidi he phudng trinh
I. Ax 2y
ir + TT^^ +y • I 4.T 2?y
V i d u 5. Til / -i- + y = ^ I (2; + j/)^ - 3 I J - y = 1 \r -- jy)-^ = 1
{ 3 = + bx'^y + 10x3j/2 + I0x2y3 + 5x^4 ^ ^5 ( j ) 1 = x^ - Sx^y + 10x3y2 _ iQ^2y3 ^ 53.^4 _ ^5 (2)
Cong (1) ua (2), lay (1) irtif (2) </ieo ve ta duac
2 = + lOx^vl + 5xy^ ^ f - = x " + 10x2y2 + 5y4 (3) 1 = 5x4y + 10x2y3 + y5 ! ^ 5^4 ^ i0x2,y2 + (4)
\ Lay (4) friif ( 3 ) , lay (3) c p n ^ ( 4 ) , ta duac
1 - ^ = 4^4 _ 4,^4 y .r
- + - = 2 (3x^ + 10x2y2 + 3y4)
X y ^ ^
Ta duac bai toan sau.
^ + I = (3x2 + y2) ( , 2 + 3 , 2 ) .
B a i t o a n 156 ( C h o n dpi t u y e n t p H o C h i M i n h duT t h i H S G quoc gia n a m h o c 2 0 0 2 - 2 0 0 3 ) . Gidi he phuang trinh
B a i t o a n 157 ( D e n g h i O l y m p i c 3 0 / 0 4 / 2 0 1 1 ) . Gidi he phuang tnnh
(3x + y ) ( x + 3 y ) = 14 ( x + y ) (x2 + 14xy + y2) = 36.
(5iai- D i e u kien x y > 0. D a t u = y/x >0, v = ^/y > 0. T h a y vao he d a cho r uv (3u2 + 02) (3t;2 + u2) = 14 j
\2 + v^) {u^ + Uu'^v^ + v^) = 36
f ut; (3w4 + 101x2^2 + 3t,4) = 14 ằ w?-""
\2 + 15u2i;4 + t;6 = 36
/ 6 u S + 20uV + 6ut;^ = 28 (1) '
\ô + 1 5 u V + 15u2i;'ằ + 1,6 = 36. (2)
Cong (1) v a (2), trir (1) v a (2) t a ditcJc ' , . / + 6u^v + 15u^t'2 + 2Qu^v^ + 15u2(;4 + 6uv^ + = 64
I u** - 6u^t' + 15u'*i;2 ^ 20u^v^ + 15u2t;4 _ 6ut;^ + v'^ = 8 / (u + ô)'5 = 26 u + D = 2
u - u
Vay (u;w) = 1 - . C a c nghiem
c u a h e l a (x; y) = + \/2; ^ - v/2) , (x; y) = Q - v ^ ; ^ +
V i d u 6. 7 ^ { ^ + y Z p 4=. { + ^js I 1 to CO
f 5 = x^ + 7x6y + 21x5y2 + 35x4y3 + SSx^y^ + 21x2y^''' + 7xy^ + y^ (1)
\ = x^ - 7x^y + 21x^2 _ 353.4,3 + 353.3,4 _ 213-2,5 ^ 73.^0 _ , 7 (2)
Cqng (1) (2), lay (2) <r-?jf (1) theo ve ta duac
3 = x^ + 21x^y2 + SSx^y" + 7xy*' 2 = 7x*'y + 3 5 x V + 21x2y'^ + y^
- = . T 6 + 2 1. T V + 35.T2y4 + 7y6
5
- = 7x'' + 35x4y2 + 21x2y4 + y^.
y , ^ J, ; Cpn^ i;d fni hai phuang trinh cuoi ta duac
2 3
X y 2 3
6x6 + 143^^4,2 _ 143,2,2 _ g , 6 2 (4x6 + 28x''y2 + 28x2y'' + 4y^)
2 (3x6 + 7^.4,2 _ 73^^2,4 _ 3 , 6 ) 1, J u;
i + f = 8 (x6 + 7x4y2 + 7x2y4 + , 6 )
2 _ 3 ^ 2 (x2 - y2) (3.^4 ^ 10^2 2 ^ 3 4 )
£ + 2 ^ 8( x 2 + 2;2)(^4 + 6 x V + / ) . .
a; y / Ta dt^pfc 6di fodn sau.
Bai toan 158. Gidi h$
- - ^ = 2 (x2 - y2) (3^4 + io^2y2 + 3^4)
X y '
Bai toan 159 (De nghi Olympic 30/03/2011). Gidi h$ phMng trinh 4 4 121a- - 122y (1)
•-^ - y =
. „ „ 122x4- 121y x4 + 14xV + 2/^ = x2 + y2 0 , 2 - (2)
Giai. Dieu kicn x 0, y ^.^ 0. Neu x = ± y thi (1) vo li. Xet x y va x -y (1) <^ 4xy (x^ - y^) = 121x - 122y.
(2) ^ (x'' + 14x2y2 + y4) + y^) = 122x + 121y.
Lan litot nhan (3) va (4) cho (x + y), (x - y) ta dudc 4xy (x* - y^) (x + y) = (121x - 122y) (x + y ) .
^ (x" + 14x2y2 + y4) (x2 + y2) (x - y) = (122x + 121y) (x - y ) . De y rang
(122x + 121y) (x - y) - (121x - 122y) (x + y)
= (122x2 - xy - 121y2) - (121x2 _ ,j.y _ ^22y'^) = + y^.
Do do lay (6) tnr (5) theo ve ta diWc
(x^ + 14x2y2 + y4) (x2 + y2) (x - y) - 4xy (x^ - y^) (x + y) = x^ + y'
^ (x^ + 14x2y2 + y4) (x - y) - 4xy (x^ - y^) (x + y) = 1 (x - y) [(x^ + 14x2y2 + i/) - 4xy (x^ + 2xy + y^)] = 1
^{x-y) [(x^ + 14x2y2 + y^) - 4x^y - Sx^y^ - 4xy=*] = 1
^ (x - y) [x"* - 4x^y + Gx^y^ - 4xy^ + y^] = 1 (x - y) (x - y)^ = 1 <^ (x - y)^ = 1 O X - y = 1.
Dat ( = X + y. Khi do
x^ - y2 = (x + y) (x - y) = i .
(3) (4)
(5) (6)
(x + yf + (x - y)^ 2 • i
x'^ + ?/''^ = 5 •
t - 1 '• '^'^ !
4xy = (x + y)2 - (x - yf = 1^ - 1, y = '
t - \3 - < l i i 121x - 122y - 121 (x - y) - y = 121 - — = . Thay vao (3) ta ditdc
(^2 ^ 1) —1~2 ^ ^ ^5 „ J ^ 243 - t ^ <^ = 243 ^ i = 3.
Vay l ^ + ^ ^ l < ^ { ^ = J Thii lai thay thoa man. i ,, 1.11.3 Su* dung bat dang thiJc lu'dng giac trong t a m giac d6 sang
tao v a xay dtftig thuat giai phirdng trinh Itfdng giac hai an.
Trong de thi vao Dai hoc Nong nghiep 1, nam 1995 co bai toan sau.
Bai toan 160. Giai phuang trinh
sin2x + sin2y + sin2(x + y) = ^. (1) PhUdng trinh nay khien ta lien tucing den mot bat dang tlnic cd ban trong
tam giac : Vdi moi tam giac ADC ta c6 / •; >:
mi^A + Hhi^B-^sui^C<l, (2)
dau bang xay ra khi va chi khi = S = C = - . Trong ve trai cua (2), lay o
A = X, B = y, C = n - {x + y ) , ta thu ditdc ve trai ciia (1). Ldi giai cua phudng trinh (1) ciing tlm dUdc di.fa tren cd scf phep chiing minh bat dRng thiic (2).
Giai. Ta c6 ' ' , cos2x + cos2y , . 2/ , N ^
(1)<^1 +sm'(x + y) = -
9 1 - cos(x -f y) cos(x - y) + 1 - cos2(x + y) = -
<^ cos2(x + y) + cos(x - y) cos(x + y) + ^ = 0. (3)
ôm (3) la phitdng trinh bac hai theo cos(x + y), ta c6 A = cos2(x - y) - 1 < 0.
Vi cos^(x - y ) < 1 nen de phudng trinh c6 nghiem thi cos^(x - y ) = 1. Do CJQ cos(a; - y) — 1
cos{x + y) = -
cos(x - y ) = - 1
cos(x - y )
cos{x + ?/) = -cos(x — y) ,2
1 2
(4)
Ta C O
X — y = k2-n 2TT
X - y = k2n
x + y = ± — +12TT 2n
X - y = TT + fc27r o
x + y = ± ^ +/27r. (^)
TT
X — y = TT + fc27r x + y = ^ + /27r
X — y = TT + k2n
X + y =
X = — + fcTT + Z T T y = - + ln - kn
TT
X = — — + A;7r + / T T y = - - +1-K - k-K.
2-K , X = — + kn + In
y = - - + ln - kn
TT