Solve for r (complex solution)
\left\{\begin{matrix}r=-\frac{3\left(\pi \beta \right)^{-\frac{1}{2}}\sqrt{Q}}{2}\text{; }r=\frac{3\left(\pi \beta \right)^{-\frac{1}{2}}\sqrt{Q}}{2}\text{, }&Q\neq 0\text{ and }\beta \neq 0\\r\neq 0\text{, }&Q=0\text{ and }\beta =0\end{matrix}\right.
Solve for Q
Q=\frac{4\pi \beta r^{2}}{9}
r\neq 0
Solve for r
\left\{\begin{matrix}r=\frac{3\sqrt{\frac{Q}{\pi \beta }}}{2}\text{; }r=-\frac{3\sqrt{\frac{Q}{\pi \beta }}}{2}\text{, }&\left(\beta <0\text{ and }Q<0\right)\text{ or }\left(\beta >0\text{ and }Q>0\right)\\r\neq 0\text{, }&Q=0\text{ and }\beta =0\end{matrix}\right.
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