解方程x四方-x²+2xy+y²+1=0 x²+2xy+6x+2y²+4y+10=0x四方-x²+2xy+y²+1=0x²+2xy+6x+2y²+4y+10=0

Question

解方程x四方-x²+2xy+y²+1=0 x²+2xy+6x+2y²+4y+10=0
x四方-x²+2xy+y²+1=0
x²+2xy+6x+2y²+4y+10=0

Answer 1

x^4-x^2+2xy+y^2+1=0
x^4-2x^2+1+x^2+2xy+y^2=0
(x^2-1)^2+(x+y)^2=0
因为(x^2-1)^2>=0,(x+y)^2>=0
当且仅当(x^2-1)^2=0,(x+y)^2=0等式才成立
x=±1,y=-x
所以x=1,y=-1或x=-1,y=1

x²+2xy+6x+2y²+4y+10=0
x²+2xy+y²+y²-2y+1+6x+6y+9=0
(x+y)^2+(y-1)^2+6(x+y)+9=0
(x+y)^2+6(x+y)+9+(y-1)^2=0
(x+y+3)^2+(y-1)^2=0
因为(x+y+3)^2>=0,(y-1)^2>=0
当且仅当(x+y+3)^2=0,(y-1)^2=0等式才成立
即y=1,x+y+3=0
x+1+3=0
x=-4
所以x=-4,y=1