Solve F=mg+mv^2/2

$$F = m g + \frac { m v ^ { 2 } } { 2 }$$

Solve for g (complex solution)

$$\left\{\begin{matrix}g=-\frac{v^{2}}{2}+\frac{F}{m}\text{, }&m\neq 0\\g\in \mathrm{C}\text{, }&F=0\text{ and }m=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$mg+\frac{mv^{2}}{2}=F$$

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

$$mg=F-\frac{mv^{2}}{2}$$

Multiply both sides of the equation by 2.

$$2mg=2F-mv^{2}$$

Divide both sides by 2m.

$$\frac{2mg}{2m}=\frac{2F-mv^{2}}{2m}$$

Dividing by 2m undoes the multiplication by 2m.

$$g=\frac{2F-mv^{2}}{2m}$$

Divide -mv^{2}+2F by 2m.

$$g=-\frac{v^{2}}{2}+\frac{F}{m}$$

Solve for g

$$\left\{\begin{matrix}g=-\frac{v^{2}}{2}+\frac{F}{m}\text{, }&m\neq 0\\g\in \mathrm{R}\text{, }&F=0\text{ and }m=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$mg+\frac{mv^{2}}{2}=F$$

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

$$mg=F-\frac{mv^{2}}{2}$$

Multiply both sides of the equation by 2.

$$2mg=2F-mv^{2}$$

Divide both sides by 2m.

$$\frac{2mg}{2m}=\frac{2F-mv^{2}}{2m}$$

Dividing by 2m undoes the multiplication by 2m.

$$g=\frac{2F-mv^{2}}{2m}$$

Divide -mv^{2}+2F by 2m.

$$g=-\frac{v^{2}}{2}+\frac{F}{m}$$

Solve for F

$$F=\frac{m\left(v^{2}+2g\right)}{2}$$