Solve w=1/2m(v_2^2-v_1^2)

$$w = \frac { 1 } { 2 } m ( v _ { 2 } ^ { 2 } - v _ { 1 } ^ { 2 } )$$

Solve for m (complex solution)

$$\left\{\begin{matrix}m=\frac{2w}{v_{2}^{2}-v_{1}^{2}}\text{, }&v_{2}\neq v_{1}\text{ and }v_{2}\neq -v_{1}\\m\in \mathrm{C}\text{, }&\left(v_{2}=-v_{1}\text{ or }v_{2}=v_{1}\right)\text{ and }w=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply \frac{1}{2}m by v_{2}^{2}-v_{1}^{2}.

$$w=\frac{1}{2}mv_{2}^{2}-\frac{1}{2}v_{1}^{2}m$$

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

$$\frac{1}{2}mv_{2}^{2}-\frac{1}{2}v_{1}^{2}m=w$$

Combine all terms containing m.

$$\left(\frac{1}{2}v_{2}^{2}-\frac{1}{2}v_{1}^{2}\right)m=w$$

The equation is in standard form.

$$\frac{v_{2}^{2}-v_{1}^{2}}{2}m=w$$

Divide both sides by \frac{1}{2}v_{2}^{2}-\frac{1}{2}v_{1}^{2}.

$$\frac{2\times \frac{v_{2}^{2}-v_{1}^{2}}{2}m}{v_{2}^{2}-v_{1}^{2}}=\frac{2w}{v_{2}^{2}-v_{1}^{2}}$$

Dividing by \frac{1}{2}v_{2}^{2}-\frac{1}{2}v_{1}^{2} undoes the multiplication by \frac{1}{2}v_{2}^{2}-\frac{1}{2}v_{1}^{2}.

$$m=\frac{2w}{v_{2}^{2}-v_{1}^{2}}$$

Solve for m

$$\left\{\begin{matrix}m=\frac{2w}{v_{2}^{2}-v_{1}^{2}}\text{, }&|v_{2}|\neq |v_{1}|\\m\in \mathrm{R}\text{, }&w=0\text{ and }|v_{2}|=|v_{1}|\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply \frac{1}{2}m by v_{2}^{2}-v_{1}^{2}.

$$w=\frac{1}{2}mv_{2}^{2}-\frac{1}{2}v_{1}^{2}m$$

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

$$\frac{1}{2}mv_{2}^{2}-\frac{1}{2}v_{1}^{2}m=w$$

Combine all terms containing m.

$$\left(\frac{1}{2}v_{2}^{2}-\frac{1}{2}v_{1}^{2}\right)m=w$$

The equation is in standard form.

$$\frac{v_{2}^{2}-v_{1}^{2}}{2}m=w$$

Divide both sides by \frac{1}{2}v_{2}^{2}-\frac{1}{2}v_{1}^{2}.

$$\frac{2\times \frac{v_{2}^{2}-v_{1}^{2}}{2}m}{v_{2}^{2}-v_{1}^{2}}=\frac{2w}{v_{2}^{2}-v_{1}^{2}}$$

Dividing by \frac{1}{2}v_{2}^{2}-\frac{1}{2}v_{1}^{2} undoes the multiplication by \frac{1}{2}v_{2}^{2}-\frac{1}{2}v_{1}^{2}.

$$m=\frac{2w}{v_{2}^{2}-v_{1}^{2}}$$

Solve for v_1 (complex solution)

$$\left\{\begin{matrix}v_{1}=-\sqrt{v_{2}^{2}-\frac{2w}{m}}\text{; }v_{1}=\sqrt{v_{2}^{2}-\frac{2w}{m}}\text{, }&m\neq 0\\v_{1}\in \mathrm{C}\text{, }&w=0\text{ and }m=0\end{matrix}\right.$$

Solve for v_1

$$\left\{\begin{matrix}v_{1}=\sqrt{v_{2}^{2}-\frac{2w}{m}}\text{; }v_{1}=-\sqrt{v_{2}^{2}-\frac{2w}{m}}\text{, }&\left(m>0\text{ or }w\geq \frac{mv_{2}^{2}}{2}\right)\text{ and }\left(m<0\text{ or }w\leq \frac{mv_{2}^{2}}{2}\right)\text{ and }m\neq 0\\v_{1}\in \mathrm{R}\text{, }&w=0\text{ and }m=0\end{matrix}\right.$$