- #1

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## Homework Statement

Find all real numbers ##x## such that,

[tex]\left(x-\frac{1}{x}\right)^{1/2}+\left(1-\frac{1}{x}\right)^{1/2}=x[/tex]

## Homework Equations

## The Attempt at a Solution

Moving the second term on LHS to the right and squaring both the sides,

[tex]x-\frac{1}{x}=x^2+1-\frac{1}{x}-2x\sqrt{1-\frac{1}{x}}[/tex]

Rearranging,

[tex]x^2-x+1=2x\sqrt{1-\frac{1}{x}}[/tex]

Squaring again leads to terms containing 4th and 3rd power of x and I don't find a way to proceed further. Here's the expression I get after squaring and rearranging,

[tex]x^4+2x^3-x^2-2x+1=0[/tex]

Any help is appreciated. Thanks!