Question

Solve the following equation:
$\sqrt{\frac{2x}{x+1}}-\sqrt{\frac{2(x+1)}{x}}=1$

Answer 1

$\sqrt{\frac{2x}{x+1}}-\sqrt{\frac{2(x+1)}{x}}=1$

Let $t=\sqrt{\frac{x}{x+1}}$

$\sqrt{2}(t-\frac{1}{t})=1⟹\sqrt{2}{t}^2-t-\sqrt{2}=0⟹t=\sqrt{2},-\frac{1}{\sqrt{2}}$

$⟹\sqrt{\frac{x}{x+1}}=\sqrt{2},-\frac{1}{\sqrt{2}}⟹x=1,-2$