若实数xy满足等式xˆ2=y+2yˆ2=x+2(x≠y),则xˆ3-2xy+yˆ3的值

Question

若实数xy满足等式xˆ2=y+2 yˆ2=x+2(x≠y),则xˆ3-2xy+yˆ3的值

Answer 1

相减
x^2-y^2=y-x
(x+y)(x-y)=-(x-y)
x≠y
两边除以x-y
x+y=-1
相加
x^2+y^2=x+y+4=-1+4=3
(x+y)^2=x^2+2xy+y^2=1
所以xy=(1-x^2-y^2)/2=-1
所以原式=(x^3+y^3)-2xy
=(x+y)(x^2-xy+y^2)-2xy
=-(x^2-xy+y^2)-2xy
=-(x^2+y^2)-xy
=-3-(-1)
=-2