Solve W=1/2mV_f^2-1/2mV_0^2

$$W = \frac { 1 } { 2 } m V _ { f } ^ { 2 } - \frac { 1 } { 2 } m V _ { 0 } ^ { 2 }$$

Solve for V_0 (complex solution)

$$\left\{\begin{matrix}V_{0}=-\sqrt{V_{f}^{2}-\frac{2W}{m}}\text{; }V_{0}=\sqrt{V_{f}^{2}-\frac{2W}{m}}\text{, }&m\neq 0\\V_{0}\in \mathrm{C}\text{, }&W=0\text{ and }m=0\end{matrix}\right.$$

Solve for V_f (complex solution)

$$\left\{\begin{matrix}V_{f}=-\sqrt{V_{0}^{2}+\frac{2W}{m}}\text{; }V_{f}=\sqrt{V_{0}^{2}+\frac{2W}{m}}\text{, }&m\neq 0\\V_{f}\in \mathrm{C}\text{, }&W=0\text{ and }m=0\end{matrix}\right.$$

Solve for V_0

$$\left\{\begin{matrix}V_{0}=\sqrt{V_{f}^{2}-\frac{2W}{m}}\text{; }V_{0}=-\sqrt{V_{f}^{2}-\frac{2W}{m}}\text{, }&\left(m>0\text{ or }W\geq \frac{mV_{f}^{2}}{2}\right)\text{ and }\left(m<0\text{ or }W\leq \frac{mV_{f}^{2}}{2}\right)\text{ and }m\neq 0\\V_{0}\in \mathrm{R}\text{, }&W=0\text{ and }m=0\end{matrix}\right.$$

Solve for V_f

$$\left\{\begin{matrix}V_{f}=\sqrt{V_{0}^{2}+\frac{2W}{m}}\text{; }V_{f}=-\sqrt{V_{0}^{2}+\frac{2W}{m}}\text{, }&\left(m<0\text{ or }W\geq -\frac{mV_{0}^{2}}{2}\right)\text{ and }\left(m>0\text{ or }W\leq -\frac{mV_{0}^{2}}{2}\right)\text{ and }m\neq 0\\V_{f}\in \mathrm{R}\text{, }&W=0\text{ and }m=0\end{matrix}\right.$$