Solve 2ax=Vf^2-V_1^2

$$2 a x = V f ^ { 2 } - V _ { 1 } ^ { 2 }$$

Solve for V (complex solution)

$$\left\{\begin{matrix}V=\frac{2ax+V_{1}^{2}}{f^{2}}\text{, }&f\neq 0\\V\in \mathrm{C}\text{, }&\left(a=-\frac{V_{1}^{2}}{2x}\text{ and }x\neq 0\text{ and }f=0\right)\text{ or }\left(x=0\text{ and }V_{1}=0\text{ and }f=0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Swap sides so that all variable terms are on the left hand side.

$$Vf^{2}-V_{1}^{2}=2ax$$

Add V_{1}^{2} to both sides.

$$Vf^{2}=2ax+V_{1}^{2}$$

The equation is in standard form.

$$f^{2}V=2ax+V_{1}^{2}$$

Divide both sides by f^{2}.

$$\frac{f^{2}V}{f^{2}}=\frac{2ax+V_{1}^{2}}{f^{2}}$$

Dividing by f^{2} undoes the multiplication by f^{2}.

$$V=\frac{2ax+V_{1}^{2}}{f^{2}}$$

Solve for V

$$\left\{\begin{matrix}V=\frac{2ax+V_{1}^{2}}{f^{2}}\text{, }&f\neq 0\\V\in \mathrm{R}\text{, }&\left(a=-\frac{V_{1}^{2}}{2x}\text{ and }x\neq 0\text{ and }f=0\right)\text{ or }\left(x=0\text{ and }V_{1}=0\text{ and }f=0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Swap sides so that all variable terms are on the left hand side.

$$Vf^{2}-V_{1}^{2}=2ax$$

Add V_{1}^{2} to both sides.

$$Vf^{2}=2ax+V_{1}^{2}$$

The equation is in standard form.

$$f^{2}V=2ax+V_{1}^{2}$$

Divide both sides by f^{2}.

$$\frac{f^{2}V}{f^{2}}=\frac{2ax+V_{1}^{2}}{f^{2}}$$

Dividing by f^{2} undoes the multiplication by f^{2}.

$$V=\frac{2ax+V_{1}^{2}}{f^{2}}$$

Solve for V_1 (complex solution)

$$V_{1}=-\sqrt{Vf^{2}-2ax}$$

$$V_{1}=\sqrt{Vf^{2}-2ax}$$

Solve for V_1

$$V_{1}=\sqrt{Vf^{2}-2ax}$$

$$V_{1}=-\sqrt{Vf^{2}-2ax}\text{, }V\geq \frac{2ax}{f^{2}}\text{ or }f=0$$