Solve V_{s}=sqrt{vR^2+vL^2}

$$V _ { s } = \sqrt { v R ^ { 2 } + v L ^ { 2 } }$$

Solve for L (complex solution)

$$\left\{\begin{matrix}L=-\sqrt{-R^{2}+\frac{V_{s}^{2}}{v}}\text{; }L=\sqrt{-R^{2}+\frac{V_{s}^{2}}{v}}\text{, }&v\neq 0\text{ and }\left(V_{s}=0\text{ or }arg(V_{s})<\pi \right)\\L\in \mathrm{C}\text{, }&V_{s}=0\text{ and }v=0\end{matrix}\right.$$

Solve for R (complex solution)

$$\left\{\begin{matrix}R=-\sqrt{-L^{2}+\frac{V_{s}^{2}}{v}}\text{; }R=\sqrt{-L^{2}+\frac{V_{s}^{2}}{v}}\text{, }&v\neq 0\text{ and }\left(V_{s}=0\text{ or }arg(V_{s})<\pi \right)\\R\in \mathrm{C}\text{, }&V_{s}=0\text{ and }v=0\end{matrix}\right.$$

Solve for L

$$\left\{\begin{matrix}L=\sqrt{-R^{2}+\frac{V_{s}^{2}}{v}}\text{; }L=-\sqrt{-R^{2}+\frac{V_{s}^{2}}{v}}\text{, }&\left(v\neq 0\text{ and }V_{s}=0\text{ and }R=0\right)\text{ or }\left(v>0\text{ and }V_{s}\geq \sqrt{vR^{2}}\right)\\L\in \mathrm{R}\text{, }&v=0\text{ and }V_{s}=0\end{matrix}\right.$$

Solve for R

$$\left\{\begin{matrix}R=\sqrt{-L^{2}+\frac{V_{s}^{2}}{v}}\text{; }R=-\sqrt{-L^{2}+\frac{V_{s}^{2}}{v}}\text{, }&\left(v\neq 0\text{ and }V_{s}=0\text{ and }L=0\right)\text{ or }\left(v>0\text{ and }V_{s}\geq \sqrt{vL^{2}}\right)\\R\in \mathrm{R}\text{, }&v=0\text{ and }V_{s}=0\end{matrix}\right.$$