先化简,后求值：x-y/x-2y+x²-y²/x²-4xy+4y².其中x=1+√2,y=1-√2.

题目

先化简,后求值：x-y/x-2y+x²-y²/x²-4xy+4y².其中x=1+√2,y=1-√2.
要过程。。

答案

(x-y)/(x-2y)+(x²-y²)/(x²-4xy+4y²)
=(x-y)(x-2y)/(x-2y)²+(x²-y²)/(x²-4xy+4y²)
=(x²-3xy+2y²+x²-y²)/(x²-4xy+4y²)
=(2x-y)(x-y)/(x-2y)²
∵x=1+√2,y=1-√2
∴x-y=2√2 2x-y=1+3√2 x-2y=3√2-1
∴(x-y)/(x-2y)+(x²-y²)/(x²-4xy+4y²)=2√2(1+3√2)/(3√2-1)=(24+38√2)/17.