Solve E=1/2mv^2+mgh,quadv

$$E = \frac { 1 } { 2 } m v ^ { 2 } + m g h , \quad v$$

Solve for g (complex solution)

$$\left\{\begin{matrix}g=\frac{2E-mv^{2}}{2hm}\text{, }&h\neq 0\text{ and }m\neq 0\\g\in \mathrm{C}\text{, }&\left(E=\frac{mv^{2}}{2}\text{ and }h=0\right)\text{ or }\left(E=0\text{ and }m=0\text{ and }h\neq 0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$\frac{1}{2}mv^{2}+mgh=E$$

Subtract \frac{1}{2}mv^{2} from both sides.

$$mgh=E-\frac{1}{2}mv^{2}$$

The equation is in standard form.

$$hmg=-\frac{mv^{2}}{2}+E$$

Divide both sides by mh.

$$\frac{hmg}{hm}=\frac{-\frac{mv^{2}}{2}+E}{hm}$$

Dividing by mh undoes the multiplication by mh.

$$g=\frac{-\frac{mv^{2}}{2}+E}{hm}$$

Solve for g

$$\left\{\begin{matrix}g=\frac{2E-mv^{2}}{2hm}\text{, }&h\neq 0\text{ and }m\neq 0\\g\in \mathrm{R}\text{, }&\left(E=\frac{mv^{2}}{2}\text{ and }h=0\right)\text{ or }\left(E=0\text{ and }m=0\text{ and }h\neq 0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$\frac{1}{2}mv^{2}+mgh=E$$

Subtract \frac{1}{2}mv^{2} from both sides.

$$mgh=E-\frac{1}{2}mv^{2}$$

The equation is in standard form.

$$hmg=-\frac{mv^{2}}{2}+E$$

Divide both sides by mh.

$$\frac{hmg}{hm}=\frac{-\frac{mv^{2}}{2}+E}{hm}$$

Dividing by mh undoes the multiplication by mh.

$$g=\frac{-\frac{mv^{2}}{2}+E}{hm}$$

Solve for E

$$E=\frac{m\left(v^{2}+2gh\right)}{2}$$