Solve sqrt{x}+sqrt{{left(w/rright)}^2x}=q

$$\sqrt{ x } + \sqrt{ { \left( \frac{ w }{ r } \right) }^{ 2 } x } = q$$

Solve for q

$$q=\frac{\sqrt{x}\left(|r|+|w|\right)}{|r|}$$

$$x\geq 0\text{ and }r\neq 0$$

Solve for r

$$\left\{\begin{matrix}r=-\frac{\sqrt{x}w}{q-\sqrt{x}}\text{; }r=\frac{\sqrt{x}w}{q-\sqrt{x}}\text{, }&w\neq 0\text{ and }x\neq q^{2}\text{ and }q-\sqrt{x}\geq 0\text{ and }x>0\\r\neq 0\text{, }&\left(q=0\text{ and }x=0\right)\text{ or }\left(w=0\text{ and }x=q^{2}\text{ and }q>0\right)\end{matrix}\right.$$