<s>[INST] Real value of equation $(x-\frac{1}{x})^\frac{1}{2}+(1-\frac{1}{x})^\frac{1}{2}=x$ Find the real value of x in the equation $(x-\frac{1}{x})^\frac{1}{2}+(1-\frac{1}{x})^\frac{1}{2}=x$
I tried to square the whole term and after expansion not getting the result.
 [/INST] We may assume $x>0$. From $\sqrt{x^2-1}+\sqrt{x-1}=x\sqrt{x}$, subtract $\sqrt{x-1}$ from both side and squaring gives us $$x^2-1=x^3+x-1-2x\sqrt{x^2-x}$$
Simplifying, $$(x^2-x)-2\sqrt{x^2-x}+1=0$$
Can you go on from here?
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