Solve for t h=-16t^2+80t+96

$h = - 16 t^{2} + 80 t + 96$

Step 1

Rewrite the equation as $- 16 t^{2} + 80 t + 96 = h$ .

$- 16 t^{2} + 80 t + 96 = h$

Step 2

Subtract $h$ from both sides of the equation.

$- 16 t^{2} + 80 t + 96 - h = 0$

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

Substitute the values $a = - 16$ , $b = 80$ , and $c = 96 - h$ into the quadratic formula and solve for $t$ .

$\frac{- 80 \pm \sqrt{80^{2} - 4 \cdot (- 16 \cdot (96 - h))}}{2 \cdot - 16}$

Step 5

Simplify.

Simplify the numerator.

Raise $80$ to the power of $2$ .

$t = \frac{- 80 \pm \sqrt{6400 - 4 \cdot - 16 \cdot (96 - h)}}{2 \cdot - 16}$

Multiply $- 4$ by $- 16$ .

$t = \frac{- 80 \pm \sqrt{6400 + 64 \cdot (96 - h)}}{2 \cdot - 16}$

Apply the distributive property.

$t = \frac{- 80 \pm \sqrt{6400 + 64 \cdot 96 + 64 (- h)}}{2 \cdot - 16}$

Multiply $64$ by $96$ .

$t = \frac{- 80 \pm \sqrt{6400 + 6144 + 64 (- h)}}{2 \cdot - 16}$

Multiply $- 1$ by $64$ .

$t = \frac{- 80 \pm \sqrt{6400 + 6144 - 64 h}}{2 \cdot - 16}$

Add $6400$ and $6144$ .

$t = \frac{- 80 \pm \sqrt{12544 - 64 h}}{2 \cdot - 16}$

Factor $64$ out of $12544 - 64 h$ .

Factor $64$ out of $12544$ .

$t = \frac{- 80 \pm \sqrt{64 (196) - 64 h}}{2 \cdot - 16}$

Factor $64$ out of $- 64 h$ .

$t = \frac{- 80 \pm \sqrt{64 (196) + 64 (- h)}}{2 \cdot - 16}$

Factor $64$ out of $64 (196) + 64 (- h)$ .

$t = \frac{- 80 \pm \sqrt{64 (196 - h)}}{2 \cdot - 16}$

Rewrite $64 (196 - h)$ as $8^{2} (14^{2} - h)$ .

Rewrite $64$ as $8^{2}$ .

$t = \frac{- 80 \pm \sqrt{8^{2} (196 - h)}}{2 \cdot - 16}$

Rewrite $196$ as $14^{2}$ .

$t = \frac{- 80 \pm \sqrt{8^{2} (14^{2} - h)}}{2 \cdot - 16}$

Pull terms out from under the radical.

$t = \frac{- 80 \pm 8 \sqrt{14^{2} - h}}{2 \cdot - 16}$

Raise $14$ to the power of $2$ .

$t = \frac{- 80 \pm 8 \sqrt{196 - h}}{2 \cdot - 16}$

Multiply $2$ by $- 16$ .

$t = \frac{- 80 \pm 8 \sqrt{196 - h}}{- 32}$

Simplify $\frac{- 80 \pm 8 \sqrt{196 - h}}{- 32}$ .

$t = \frac{10 \pm \sqrt{196 - h}}{4}$

Step 6

The final answer is the combination of both solutions.

$t = \frac{10 + \sqrt{196 - h}}{4}$

$t = \frac{10 - \sqrt{196 - h}}{4}$

Solve for t h=-16t^2+80t+96

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