Question

已知x+x^(-1)=3，求下列各值：

1.
x^(1/2)+x^(-1)

2.
x^2+x^(-2)

3.x^2-x^(-2)

一定要有过程。前两道可以略写但最后一道要详细的。谢谢。

Answer 1

1.[x^(1/2)+x^(-1/2)]^2=x+1/x+2=3+2=5
则
x^(1/2)+x^(-1/2)=根号5或-根号5
2.(x+1/x)^2=x^2+1/x^2+2=9
则
x^2+1/x^2=7
3.(x-1/x)^2=(x^2+1/x^2)-2=7-2=5
则
x-1/x=根号5或-根号5

x^2-1/x^2=(x+1/x)(x-1/x)
=3*(根号5)或-3(根号5)

Answer 2

（x+1/x）²＝9＝x²+1/x²+2.x²+1/x²＝7
（x-1/x）²＝x²+1/x²-2＝7-2＝5
x-1/x＝±√5
①x-1/x＝√5
x＝（3+√5）/2.1/x＝（3-√5）/2
x^(1/2)+x^(-1)＝√[（3+√5）/2]+（3-√5）/2
x^2+x^(-2)＝7
x^2-x^(-2)＝6√5.
②x-1/x＝-√5
x＝（3-√5）/2.1/x＝（3+√5）/2
x^2+x^(-2)＝7
x^2-x^(-2)＝-6√5.