Solve v^2=v_0^2+2a(x-x_{0})

$$v ^ { 2 } = v _ { 0 } ^ { 2 } + 2 a ( x - x _ { 0 } )$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=-\frac{v_{0}^{2}-v^{2}}{2\left(x-x_{0}\right)}\text{, }&x\neq x_{0}\\a\in \mathrm{C}\text{, }&\left(v=-v_{0}\text{ or }v=v_{0}\right)\text{ and }x=x_{0}\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply 2a by x-x_{0}.

$$v^{2}=v_{0}^{2}+2ax-2ax_{0}$$

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

$$v_{0}^{2}+2ax-2ax_{0}=v^{2}$$

Subtract v_{0}^{2} from both sides.

$$2ax-2ax_{0}=v^{2}-v_{0}^{2}$$

Combine all terms containing a.

$$\left(2x-2x_{0}\right)a=v^{2}-v_{0}^{2}$$

Divide both sides by 2x-2x_{0}.

$$\frac{\left(2x-2x_{0}\right)a}{2x-2x_{0}}=\frac{\left(v-v_{0}\right)\left(v+v_{0}\right)}{2x-2x_{0}}$$

Dividing by 2x-2x_{0} undoes the multiplication by 2x-2x_{0}.

$$a=\frac{\left(v-v_{0}\right)\left(v+v_{0}\right)}{2x-2x_{0}}$$

Divide \left(v-v_{0}\right)\left(v+v_{0}\right) by 2x-2x_{0}.

$$a=\frac{\left(v-v_{0}\right)\left(v+v_{0}\right)}{2\left(x-x_{0}\right)}$$

Solve for a

$$\left\{\begin{matrix}a=-\frac{v_{0}^{2}-v^{2}}{2\left(x-x_{0}\right)}\text{, }&x\neq x_{0}\\a\in \mathrm{R}\text{, }&x=x_{0}\text{ and }|v|=|v_{0}|\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply 2a by x-x_{0}.

$$v^{2}=v_{0}^{2}+2ax-2ax_{0}$$

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

$$v_{0}^{2}+2ax-2ax_{0}=v^{2}$$

Subtract v_{0}^{2} from both sides.

$$2ax-2ax_{0}=v^{2}-v_{0}^{2}$$

Combine all terms containing a.

$$\left(2x-2x_{0}\right)a=v^{2}-v_{0}^{2}$$

Divide both sides by 2x-2x_{0}.

$$\frac{\left(2x-2x_{0}\right)a}{2x-2x_{0}}=\frac{\left(v-v_{0}\right)\left(v+v_{0}\right)}{2x-2x_{0}}$$

Dividing by 2x-2x_{0} undoes the multiplication by 2x-2x_{0}.

$$a=\frac{\left(v-v_{0}\right)\left(v+v_{0}\right)}{2x-2x_{0}}$$

Divide \left(v-v_{0}\right)\left(v+v_{0}\right) by 2x-2x_{0}.

$$a=\frac{\left(v-v_{0}\right)\left(v+v_{0}\right)}{2\left(x-x_{0}\right)}$$

Solve for v (complex solution)

$$v=-\sqrt{2ax-2ax_{0}+v_{0}^{2}}$$

$$v=\sqrt{2ax-2ax_{0}+v_{0}^{2}}$$