Solve s=1/2gt^2+v_{0}t.quadd

$$s = \frac { 1 } { 2 } g t ^ { 2 } + v _ { 0 } t . \quad d$$

Solve for g (complex solution)

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

Steps for Solving Linear Equation

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

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

Subtract v_{0}t from both sides.

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

The equation is in standard form.

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

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

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

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

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

Solve for g

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

Steps for Solving Linear Equation

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

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

Subtract v_{0}t from both sides.

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

The equation is in standard form.

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

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

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

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

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

Solve for s

$$s=\frac{t\left(gt+2v_{0}\right)}{2}$$