- #1
Saitama
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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!