Solve for t p(t)=5000+1000t^2

$p (t) = 5000 + 1000 t^{2}$

Subtract $1000 t^{2}$ from both sides of the equation.

$p t - 1000 t^{2} = 5000$

Subtract $5000$ from both sides of the equation.

$p t - 1000 t^{2} - 5000 = 0$

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = - 1000$ , $b = p$ , and $c = - 5000$ into the quadratic formula and solve for $t$ .

$\frac{- p \pm \sqrt{p^{2} - 4 \cdot (- 1000 \cdot - 5000)}}{2 \cdot - 1000}$

Simplify.

Simplify the numerator.

Multiply $- 4$ by $- 1000$ .

$t = \frac{- p \pm \sqrt{p^{2} + 4000 \cdot - 5000}}{2 \cdot - 1000}$

Multiply $4000$ by $- 5000$ .

$t = \frac{- p \pm \sqrt{p^{2} - 20000000}}{2 \cdot - 1000}$

Multiply $2$ by $- 1000$ .

$t = \frac{- p \pm \sqrt{p^{2} - 20000000}}{- 2000}$

Simplify $\frac{- p \pm \sqrt{p^{2} - 20000000}}{- 2000}$ .

$t = \frac{p \pm \sqrt{p^{2} - 20000000}}{2000}$

The final answer is the combination of both solutions.

$t = \frac{p + \sqrt{p^{2} - 20000000}}{2000}$

$t = \frac{p - \sqrt{p^{2} - 20000000}}{2000}$