Solve m=2u{left(frac{{p}_{2}}{{p}_{1}}right)}^frac{1{2}}{p}_{1}

$$m = 2u { \left( \frac{ { p }_{ 2 } }{ { p }_{ 1 } } \right) }^{ \frac{ 1 }{ 2 } } { p }_{ 1 }$$

Solve for m (complex solution)

$$m=2\sqrt{\frac{p_{2}}{p_{1}}}p_{1}u$$

$$p_{1}\neq 0$$

Solve for p_1 (complex solution)

$$\left\{\begin{matrix}p_{1}=\frac{\left(\frac{m}{u}\right)^{2}}{4p_{2}}\text{, }&|arg(\frac{m^{2}\sqrt{\left(\frac{p_{2}u}{m}\right)^{2}}}{p_{2}u})-arg(m)|<\pi \text{ and }m\neq 0\text{ and }u\neq 0\text{ and }p_{2}\neq 0\\p_{1}\neq 0\text{, }&\left(u=0\text{ or }p_{2}=0\right)\text{ and }m=0\end{matrix}\right.$$

Solve for m

$$m=2\sqrt{\frac{p_{2}}{p_{1}}}p_{1}u$$

$$\left(p_{2}\geq 0\text{ and }p_{1}>0\right)\text{ or }\left(p_{2}\leq 0\text{ and }p_{1}<0\right)$$

Solve for p_1

$$\left\{\begin{matrix}p_{1}=\frac{\left(\frac{m}{u}\right)^{2}}{4p_{2}}\text{, }&\left(p_{2}>0\text{ and }u<0\text{ and }m<0\right)\text{ or }\left(p_{2}>0\text{ and }u>0\text{ and }m>0\right)\text{ or }\left(p_{2}<0\text{ and }m<0\text{ and }u>0\right)\text{ or }\left(p_{2}<0\text{ and }u<0\text{ and }m>0\right)\\p_{1}\neq 0\text{, }&p_{2}=0\text{ and }m=0\\p_{1}<0\text{, }&p_{2}\leq 0\text{ and }m=0\text{ and }u=0\\p_{1}>0\text{, }&p_{2}\geq 0\text{ and }m=0\text{ and }u=0\end{matrix}\right.$$