Solve PG^2=4M^2-3PR^2

$$P G ^ { 2 } = 4 M ^ { 2 } - 3 P R ^ { 2 }$$

Solve for G (complex solution)

$$\left\{\begin{matrix}G=-\sqrt{-3R^{2}+\frac{4M^{2}}{P}}\text{; }G=\sqrt{-3R^{2}+\frac{4M^{2}}{P}}\text{, }&P\neq 0\\G\in \mathrm{C}\text{, }&M=0\text{ and }P=0\end{matrix}\right.$$

Solve for M (complex solution)

$$M=-\frac{\sqrt{P}\sqrt{G^{2}+3R^{2}}}{2}$$

$$M=\frac{\sqrt{P}\sqrt{G^{2}+3R^{2}}}{2}$$

Solve for G

$$\left\{\begin{matrix}G=\sqrt{-3R^{2}+\frac{4M^{2}}{P}}\text{; }G=-\sqrt{-3R^{2}+\frac{4M^{2}}{P}}\text{, }&\left(P\leq \frac{4\times \left(\frac{M}{R}\right)^{2}}{3}\text{ or }R=0\right)\text{ and }P>0\\G\in \mathrm{R}\text{, }&M=0\text{ and }P=0\end{matrix}\right.$$

Solve for M

$$M=\frac{\sqrt{P\left(G^{2}+3R^{2}\right)}}{2}$$

$$M=-\frac{\sqrt{P\left(G^{2}+3R^{2}\right)}}{2}\text{, }\left(G=0\text{ and }R=0\right)\text{ or }P\geq 0$$