Solve x_{0}=v_{0}t+1/2gt^2

$$x _ { 0 } = v _ { 0 } t + \frac { 1 } { 2 } g t ^ { 2 }$$

Solve for g (complex solution)

$$\left\{\begin{matrix}g=-\frac{2\left(tv_{0}-x_{0}\right)}{t^{2}}\text{, }&t\neq 0\\g\in \mathrm{C}\text{, }&x_{0}=0\text{ and }t=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$v_{0}t+\frac{1}{2}gt^{2}=x_{0}$$

Subtract v_{0}t from both sides.

$$\frac{1}{2}gt^{2}=x_{0}-v_{0}t$$

The equation is in standard form.

$$\frac{t^{2}}{2}g=x_{0}-tv_{0}$$

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

$$\frac{2\times \frac{t^{2}}{2}g}{t^{2}}=\frac{2\left(x_{0}-tv_{0}\right)}{t^{2}}$$

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

$$g=\frac{2\left(x_{0}-tv_{0}\right)}{t^{2}}$$

Solve for g

$$\left\{\begin{matrix}g=-\frac{2\left(tv_{0}-x_{0}\right)}{t^{2}}\text{, }&t\neq 0\\g\in \mathrm{R}\text{, }&x_{0}=0\text{ and }t=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$v_{0}t+\frac{1}{2}gt^{2}=x_{0}$$

Subtract v_{0}t from both sides.

$$\frac{1}{2}gt^{2}=x_{0}-v_{0}t$$

The equation is in standard form.

$$\frac{t^{2}}{2}g=x_{0}-tv_{0}$$

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

$$\frac{2\times \frac{t^{2}}{2}g}{t^{2}}=\frac{2\left(x_{0}-tv_{0}\right)}{t^{2}}$$

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

$$g=\frac{2\left(x_{0}-tv_{0}\right)}{t^{2}}$$

Solve for t (complex solution)

$$\left\{\begin{matrix}t=-\frac{\sqrt{2gx_{0}+v_{0}^{2}}+v_{0}}{g}\text{; }t=-\frac{-\sqrt{2gx_{0}+v_{0}^{2}}+v_{0}}{g}\text{, }&g\neq 0\\t=\frac{x_{0}}{v_{0}}\text{, }&g=0\text{ and }v_{0}\neq 0\\t\in \mathrm{C}\text{, }&g=0\text{ and }v_{0}=0\text{ and }x_{0}=0\end{matrix}\right.$$

Solve for t

$$\left\{\begin{matrix}t=-\frac{\sqrt{2gx_{0}+v_{0}^{2}}+v_{0}}{g}\text{; }t=-\frac{-\sqrt{2gx_{0}+v_{0}^{2}}+v_{0}}{g}\text{, }&\left(g>0\text{ or }x_{0}\leq -\frac{v_{0}^{2}}{2g}\right)\text{ and }\left(x_{0}\leq \text{Indeterminate}\text{ or }g\neq 0\right)\text{ and }\left(g<0\text{ or }\left(g\neq 0\text{ and }x_{0}\geq -\frac{v_{0}^{2}}{2g}\right)\right)\\t=\frac{x_{0}}{v_{0}}\text{, }&g=0\text{ and }v_{0}\neq 0\\t\in \mathrm{R}\text{, }&g=0\text{ and }v_{0}=0\text{ and }x_{0}=0\end{matrix}\right.$$