Question

已知X²＋X－1=0，求X[(1－2/(1－X)÷(X＋1)－(X²－1)/(X²－2X＋1)的值

Answer 1

X[(1－2/(1－X)÷(X＋1)－(X²－1)/(X²－2X＋1)]
=X[(1－2/(1－X^2)－(X+1)(X－1)/(X-1)^2]
=X[(1－2/X)－(X+1)/(X-1)]
=X-2-X(X+1)/(X-1)
=X-2-1/(X-1)

Answer 2

±√5-1

解法如下：
X²＋X－1=0
X=-(X²-1)
X/(X²-1)=-1

X²＋X-1=0
(X＋1/2)²-1/4-1=0
(X＋1/2)²=5/4
X+1/2=±√5/2
X=±√5/2-1/2
X=(±√5-1)/2

X[(1－2/(1－X)÷(X＋1)－(X²－1)/(X²－2X＋1)]
=X[1-2/(1-X²)-(X+1)(X-1)/(X-1)(X-1)]
=X[1+2/(X²-1)-(X+1)/(X-1)]
=X[1+2/(X²-1)-(X+1)²/(X²-1)]
=X(X²-1+2-X²-2X-1)/(X²-1)
=-2X²/(X²-1)
=-2X(-1)
=2X
=2(±√5-1)/2
=±√5-1