已知实数X满足x²-2x-1=0,分别求出x-1/x,x+1/x,x²+1/x²,x^4+1/x^4的值.

已知实数X满足x²-2x-1=0,分别求出x-1/x,x+1/x,x²+1/x²,x^4+1/x^4的值.
已知实数X满足x²-2x-1=0,分别求出x-1/x,x+1/x,x²+1/x²,x^4+1/x^4的值.

已知实数X满足x²-2x-1=0,分别求出x-1/x,x+1/x,x²+1/x²,x^4+1/x^4的值.
①
x²-2x-1=0
两边同时除以x得
x-2-1/x=0
x-1/x=2
②
x-1/x=2
两边同时平方得
x²-2+1/x²=4
x²+1/x²=6
③
(x+1/x)²=x²+2+1/x²=6+2=8
所以x+1/x=±2√2
④
x^4+1/x^4
=x^4+2+1/x^4-2
=(x²+1/x²)²-2
=6²-2
=36-2
=34
答案：x-1/x=2
x+1/x=±2√2
x²+1/x²=6
x^4+1/x^4=34