Solve 1/2alphat^2-v_{0}t-x_{0}=0

$$\frac { 1 } { 2 } \alpha t ^ { 2 } - v _ { 0 } t - x _ { 0 } = 0$$

Solve for v_0 (complex solution)

$$\left\{\begin{matrix}v_{0}=\frac{t\alpha }{2}-\frac{x_{0}}{t}\text{, }&t\neq 0\\v_{0}\in \mathrm{C}\text{, }&x_{0}=0\text{ and }t=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract \frac{1}{2}\alpha t^{2} from both sides. Anything subtracted from zero gives its negation.

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

Add x_{0} to both sides.

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

The equation is in standard form.

$$\left(-t\right)v_{0}=-\frac{\alpha t^{2}}{2}+x_{0}$$

Divide both sides by -t.

$$\frac{\left(-t\right)v_{0}}{-t}=\frac{-\frac{\alpha t^{2}}{2}+x_{0}}{-t}$$

Dividing by -t undoes the multiplication by -t.

$$v_{0}=\frac{-\frac{\alpha t^{2}}{2}+x_{0}}{-t}$$

Divide x_{0}-\frac{\alpha t^{2}}{2} by -t.

$$v_{0}=\frac{t\alpha }{2}-\frac{x_{0}}{t}$$

Solve for v_0

$$\left\{\begin{matrix}v_{0}=\frac{t\alpha }{2}-\frac{x_{0}}{t}\text{, }&t\neq 0\\v_{0}\in \mathrm{R}\text{, }&x_{0}=0\text{ and }t=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract \frac{1}{2}\alpha t^{2} from both sides. Anything subtracted from zero gives its negation.

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

Add x_{0} to both sides.

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

The equation is in standard form.

$$\left(-t\right)v_{0}=-\frac{\alpha t^{2}}{2}+x_{0}$$

Divide both sides by -t.

$$\frac{\left(-t\right)v_{0}}{-t}=\frac{-\frac{\alpha t^{2}}{2}+x_{0}}{-t}$$

Dividing by -t undoes the multiplication by -t.

$$v_{0}=\frac{-\frac{\alpha t^{2}}{2}+x_{0}}{-t}$$

Divide x_{0}-\frac{\alpha t^{2}}{2} by -t.

$$v_{0}=\frac{t\alpha }{2}-\frac{x_{0}}{t}$$

Solve for t (complex solution)

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

Solve for t

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