x^4+x^2+1/x^4+1/x^2=4 求解
来源:百度知道 编辑:UC知道 时间:2024/06/19 17:47:39
x^4+x^2+1/x^4+1/x^2=4
x^4+1/x^4+x^2+1/x^2-4=0
(x^4+1/x^4-2)+(x^2+1/x^2-2)=0
(x^2-1/x^2)^2+(x-1/x)^2=0
所以x-1/x=0
x^2-1/x^2=0
解得x=±1
x^4+1/x^4>=2 x=+-1时取=
x^2+1/x^2>=2 x=+-1时取=
所以x=+-1时x^4+x^2+1/x^4+1/x^2=4
x^4+x^2+1/x^4+1/x^2=4
(x^2+1/x^2)^2-2+x^2+1/x^2=4
(x^2+1/x^2)^2+x^2+1/x^2-6=0
x^2+1/x^2=2或者-3(舍去)
所以x^2+1/x^2=2
(x+1/x)^2-2=2
x+1/x=+-2
x^2+-2x+1=0
x=+-1
x^4+x^2+1/x^4+1/x^2=4
(x^4-2+1/x^4)+(x^2-2+1/x^2)=0
(x^2-1/x^2)^2+(x-1/x)^2=0
所以只有当两个平方都等于0时等式才成立,因此一贯同时满足:
x^2-1/x^2=0
x-1/x=0
解得x=±1
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