. x^, 4x +3 y, Ta thay phuong trinh c6 nghỉm khi va chi khi — va
V2 y+ 7xT7 < 0, t udo suyra x=y
Thay vao (2) ta du^c: x^ - 5x^ + 14x - 4 = 6\/x^ - x + 1 . Bien doi p h u o n g trinh da cho hiong duong:
x^ +3x2 + 6 x + 4 = 8x2 _ 8 X + 8 + 3N/ 8X2 - 8X + 8 o ( x + l f+ 3( x + l) - 8 x 2 - 8 x + 8 + 3\/8x2-8x + 8 . Xet ham f (t) = t^ + 3t ta c6 f "(t) = St^ + 3 > 0. T u do ta c6: f ( x + l ) = f f^ 8 x 2 - 8 x + 8 l « x + l = \/8x2-8x + 8 o x = l ; y - l . v. / V|y h? CO nghỉm ( x ; y ) = ( l ; l ) .
KHI TRONG H f C 6 CHCTA PHUONG TRINH BAC 2
THEO AN X, HOAC y
K h i trong h? phucmg trinh c6 chua phuong trinh bac hai theo an x hoac y
ta CO the nghi deh cac huong x u ly n h u sau:
* Neu A c h i n , ta giai x theo y roi the vao p h u o n g trinh con lai cua h f de giai tiep
* Neu A khong c h i n ta thuong x u ly theo each:
+ Cpng hoac t r u cac p h u o n g trinh cua h? de tao dug-c p h u o n g trinh b^c hai c6 A chan hoac tao thanh cac hang dang thiic
+ D i i n g dieu k i ^ n A > 0 de t i m mien gia trj cua bien x,y . Sau do diing ham so de danh gia p h u o n g trinh con l ^ i tren mien gia t r i x, y vua t i m du^c:
Ta xet cac v i sau:
V i 1: G i i i cac h f p h u o n g trinh sau
a) • xy + x + y = x2 - 2 y 2 (1) a) • x 7 2 y- y V x - l = 2 x - 2 y (2) ' . ''' '\' b) • 2x2 + y2 _ 3xy + 3x - 2y + 1 = 0 4x2 - y2 + X + 4 = + y + ^ x + 4y Giii
Xet p h u o n g trinh (1) ciia h f ta c6:
xy + x + y = x 2 - 2 y 2 « x 2- x ( y + l) - 2 y 2- y = 0 . Ta coi day la phuong trinh bac 2 cua X thi ta c6: A = (y +1)2 + 8y2 + 4y = (3y +1)2 . T u do suy ra
; ^ _ y + l - ( 3 y + l ) _
2 ^ .,
X = _ y + l + (3y + l )
= 2y + l
Truang hgp 1: x = - y . T u phuong trinh (2) ciia h? ta c6 dieu kỉn: suy ra phuong trinh v6 n g h i f m
Truang hop 2: x = 2y + 1 thay vao phuong trinh t h u hai ta c6: (2y + l) V 2 y - y V2y = 2y + 2 o y 7 2 7 + 7 2 y = 2(y +1)
o (y + l) ( 7 2 y - 2) = 0 « y = 2 => X = 5 Vay h? CO mpt cap nghi^m: (x;y) = (5;2)
) Xet phuong trinh (1) ciia h? ta c6:
2x2 + y 2_ 3 x y + 3 x - 2 y + l = 0 o 2 x 2 + x ( 3 - 3 y ) + y 2- 2 y + l = 0. Coi day la phuong trinh bac 2 ciia x ta c6:
A = ( 3 - 3 y) 2- 8 ( y 2 - 2 y + l ) = y 2 - 2 y + l = ( y - i ) 2 [ x > l len: < y >0 Suy ra x_ 3 y - 3- ( y - l ) ^ y - l . X X
Truong h(?p 1: y = x +1 thay vao phuong trinh (2) ta thu du(?c:
3x2 - x + 3 = >/3x + l +V5x + 4 .,t o 3 x 2 - 3 x + (x + l- V 3 x + l ) + (x + 2 - V 5 x + 4) = 0
.(x^-x) 3 + X +1 + V3x +1 X + 2 + V5X + 4 . X +1 + V3x +1 X + 2 + V5X + 4 . 1 1 = 0 > 0 = > x ^ - x = 0<=> Do x > - - nen 3 + 7=:^ 3 X + 1 + V3X + 1 X + 2 + V5X + 4
Tmong h<?p 2: y = 2x +1 thay vao phuong trinh (2) ta thu Axxqc:
3-3x = V4x + 1 + V5x + 4 o V4x + 1 + Jsx+A + 3x-3 = 0