Question

x+y=1 & x^4+y^4=-1 then what is the value of x^2•y^2-2xy

Answer 1

There are many ways to manipulate x4+y4 out of x+y but most lead down the garden path. So we try this.

x2+y2=(x+y)2−2xy=1−2xy

and then

x3+y3=(x2+y2)(x+y)−xy(x+y)=1−3xy

and finally

x4+y4=(x3+y3)(x+y)−xy(x2+y2)=1−4xy+2x2y2=1+2(x2y2−2xy)

So

1+2(x2y2−2xy)=−1⇒x2y2−2xy=−1−12=−1