求证：（x+y-2xy）(x+y-2)+(1-xy)^2=（x+y-xy-1）^2.

来源：百度知道 编辑：UC知道 时间：2024/06/28 14:45:27

设x+y=A
（x+y-2xy）(x+y-2)+(1-xy)^2
=(A-2xy)(A-2)+(1-xy)^2
=A^2-2A-2Axy+4xy+1-2xy+x^2y^2
=A^2-2A+1-2Axy+x^2y^2
=(A-1)^2-2*A*xy+(xy)^2
=(A-1-xy)^2
=(x+y-xy-1)^2

令x+y=a，xy=b
原式化为：
（b-1）^2+(a-2)(a-2b)
=b^2-2b+1+a^2-2a-2ab+4b
=b^2+2b+1+a^2-2a-2ab
=(a^2-2ab+b^2)+2(b-a)+1
=(b-a)^2+2(b-a)+1
=(b-a+1)^2
=(xy-x-y+1)^2
=(-(xy-x-y+1))^2
=（x+y-xy-1）^2

左右都展开，自己算

左边=(x+y)^2-(2+2xy)(x+y)+4xy+1-2xy+x^2y^2
=(x+y)^2-(2+2xy)(x+y)+1+2xy+x^2y^2
=(x+y)^2-2(1+xy)(x+y)+(1+xy)^2
=[(x+y)-(1+xy)]^2
=(x+y-xy-1)^2=右边