Question

若x^2y^2-4xy+2=1-x^2-y^2,则2x+3y=

Answer 1

x²y²-4xy+2=1-x²-y²
x²y²-2xy+1+x²-2xy+y²=0
(xy-1)²+(x-y)²=0
平方项恒非负，两非负项之和=0，两非负项分别=0
x-y=0
xy-1=0
解得x=1 y=1或x=-1 y=-1
x=1 y=1时，2x+3y=2+3=5
x=-1 y=-1时，2x+3y=-2-3=-5

Answer 2

x^2y^2-4xy+2=1-x^2-y^2
两边-2xy-1
x^2y^2-2xy+1=-x^2-y^2-2xy
则 (xy-1)^2=-(x-y)^2
即 (xy-1)^2+(x-y)^2=0
∴ xy-1=0, xy=1
x-y=0, x=y
则 得
x=y=1
或
x=y=-1
(1) x=y=1, 则2x+3y=5
(2) x=y=-1, 则2x+3y=-5

Answer 3

x^2y^2-4xy+2=1-x^2-y^2
x^2y^2+x^2+y^2-4xy+1=0
2x^2y^2+2x^2+2y^2-8xy+2=0
(2x^2y^2-4xy+2)+(2x^2+2y^2-4xy)=0
2(xy-1)^2+2(x-y)^2=0
xy=1 x=y
x=y=1或x+y=-1
2x+3y=5或-5