Solve 2000=75t+at^2

$$2000 = 75t+a { t }^{ 2 }$$

Solve for a

$$a=-\frac{25\left(3t-80\right)}{t^{2}}$$

$$t\neq 0$$

Steps for Solving Linear Equation

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

$$75t+at^{2}=2000$$

Subtract 75t from both sides.

$$at^{2}=2000-75t$$

The equation is in standard form.

$$t^{2}a=2000-75t$$

Divide both sides by t^{2}.

$$\frac{t^{2}a}{t^{2}}=\frac{2000-75t}{t^{2}}$$

Dividing by t^{2} undoes the multiplication by t^{2}.

$$a=\frac{2000-75t}{t^{2}}$$

Divide 2000-75t by t^{2}.

$$a=\frac{25\left(80-3t\right)}{t^{2}}$$

Solve for t (complex solution)

$$\left\{\begin{matrix}t=-\frac{5\left(\sqrt{5\left(64a+45\right)}+15\right)}{2a}\text{; }t=-\frac{5\left(-\sqrt{5\left(64a+45\right)}+15\right)}{2a}\text{, }&a\neq 0\\t=\frac{80}{3}\text{, }&a=0\end{matrix}\right.$$

Solve for t

$$\left\{\begin{matrix}t=-\frac{5\left(\sqrt{5\left(64a+45\right)}+15\right)}{2a}\text{; }t=-\frac{5\left(-\sqrt{5\left(64a+45\right)}+15\right)}{2a}\text{, }&a\neq 0\text{ and }a\geq -\frac{45}{64}\\t=\frac{80}{3}\text{, }&a=0\end{matrix}\right.$$