Solve for P_z
P_{z}=-\frac{20-v^{2}-x}{v^{2}-20}
|v|\neq 2\sqrt{5}
Solve for v
\left\{\begin{matrix}v=\sqrt{-\frac{x+20P_{z}-20}{1-P_{z}}}\text{; }v=-\sqrt{-\frac{x+20P_{z}-20}{1-P_{z}}}\text{, }&\left(P_{z}<1\text{ or }P_{z}\geq -\frac{x}{20}+1\right)\text{ and }\left(P_{z}>1\text{ or }P_{z}\leq -\frac{x}{20}+1\right)\text{ and }\left(P_{z}=-\frac{x}{20}+1\text{ or }P_{z}\neq 1\right)\text{ and }x\neq 0\\v\in \mathrm{R}\setminus 2\sqrt{5},-2\sqrt{5}\text{, }&P_{z}=1\text{ and }x=0\end{matrix}\right.
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