解方程X²-X-4=1/X-1/X²

解方程X²-X-4=1/X-1/X²

x²-x-4=1/x-1/x²
两边同乘以x²,得

x⁴-x³-4x²-x+1=0
(x⁴-x²)+（-x³-2x²-x）+(-x²+1)=0
x²(x²-1)-x(x+1)²-(x²-1)=0
(x+1)[x²(x-1)-x(x+1)-(x-1)]=0
(x+1)[(x-1)(x²-1)-x(x+1)]=0
(x+1)(x+1)[(x-1)(x-1)-x]=0
(x+1)(x+1)(x²-3x+1)=0
(x+1)²(x²-3x+1)=0
(x+1)²=0或x²-3x+1=0
x1=x2=-1
x3=(3+√5)/2
x4=(3-√5)/2


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