Solve I=v/sqrt{R^2+x^2}

$$I = \frac { v } { \sqrt { R ^ { 2 } + x ^ { 2 } } }$$

Solve for I (complex solution)

$$I=\left(x^{2}+R^{2}\right)^{-\frac{1}{2}}v$$

$$R\neq ix\text{ and }R\neq -ix$$

Solve for R (complex solution)

$$\left\{\begin{matrix}R=-\sqrt{-x^{2}+\left(\frac{v}{I}\right)^{2}}\text{; }R=\sqrt{-x^{2}+\left(\frac{v}{I}\right)^{2}}\text{, }&|arg(\sqrt{\left(\frac{v}{I}\right)^{2}}I)-arg(v)|<\pi \text{ and }v\neq 0\text{ and }I\neq 0\\R\in \mathrm{C}\setminus -ix,ix\text{, }&I=0\text{ and }v=0\end{matrix}\right.$$

Solve for I

$$I=\frac{v}{\sqrt{x^{2}+R^{2}}}$$

$$x\neq 0\text{ or }R\neq 0$$

Solve for R

$$\left\{\begin{matrix}R=\frac{\sqrt{v^{2}-\left(Ix\right)^{2}}}{|I|}\text{; }R=-\frac{\sqrt{v^{2}-\left(Ix\right)^{2}}}{|I|}\text{, }&\left(x\neq 0\text{ and }v>0\text{ and }I>0\text{ and }I\leq \frac{v}{|x|}\right)\text{ or }\left(x\neq 0\text{ and }v<0\text{ and }I<0\text{ and }I\geq \frac{v}{|x|}\right)\text{ or }\left(I<\frac{v}{|x|}\text{ and }x\neq 0\text{ and }v>I|x|\text{ and }v>0\text{ and }I>0\right)\text{ or }\left(I>\frac{v}{|x|}\text{ and }x\neq 0\text{ and }v<I|x|\text{ and }v<0\text{ and }I<0\right)\text{ or }\left(x=0\text{ and }I>0\text{ and }v>0\right)\text{ or }\left(x=0\text{ and }I<0\text{ and }v<0\right)\\R\in \mathrm{R}\text{, }&x\neq 0\text{ and }I=0\text{ and }v=0\\R\neq 0\text{, }&I=0\text{ and }v=0\end{matrix}\right.$$