Find all real numbers such that
$$\sqrt{x-\frac{1}{x}} + \sqrt{1 – \frac{1}{x}} = x$$
My attempt to the solution :
I tried to square both sides and tried to remove the root but the equation became of 6th degree.Is there an easier method to solve this?
Best Answer
Let's multiply $$ \sqrt{x-1/x} + \sqrt{1 - 1/x} = x\tag{1} $$ by $(\sqrt{x-1/x} - \sqrt{1 - 1/x})$. Then we get $$ x-1=x(\sqrt{x-1/x} - \sqrt{1 - 1/x}) $$ $$ \sqrt{x-1/x} - \sqrt{1 - 1/x}=1-1/x\tag{2} $$ Sum up $(1)$ and $(2)$ to get $$ 2\sqrt{x-1/x}=x-1/x+1 $$ Now make the substitution $y=\sqrt{x-1/x}$. The rest is clear.