Solve for t h=-16t^2+36t+4

$h = - 16 t^{2} + 36 t + 4$

Step 1

Rewrite the equation as $- 16 t^{2} + 36 t + 4 = h$ .

$- 16 t^{2} + 36 t + 4 = h$

Step 2

Subtract $h$ from both sides of the equation.

$- 16 t^{2} + 36 t + 4 - h = 0$

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 $a = - 16$ , $b = 36$ , and $c = 4 - h$ into the quadratic formula and solve for $t$ .

$\frac{- 36 \pm \sqrt{36^{2} - 4 \cdot (- 16 \cdot (4 - h))}}{2 \cdot - 16}$

Step 5

Simplify.

Simplify the numerator.

Raise $36$ to the power of $2$ .

$t = \frac{- 36 \pm \sqrt{1296 - 4 \cdot - 16 \cdot (4 - h)}}{2 \cdot - 16}$

Multiply $- 4$ by $- 16$ .

$t = \frac{- 36 \pm \sqrt{1296 + 64 \cdot (4 - h)}}{2 \cdot - 16}$

Apply the distributive property.

$t = \frac{- 36 \pm \sqrt{1296 + 64 \cdot 4 + 64 (- h)}}{2 \cdot - 16}$

Multiply $64$ by $4$ .

$t = \frac{- 36 \pm \sqrt{1296 + 256 + 64 (- h)}}{2 \cdot - 16}$

Multiply $- 1$ by $64$ .

$t = \frac{- 36 \pm \sqrt{1296 + 256 - 64 h}}{2 \cdot - 16}$

Add $1296$ and $256$ .

$t = \frac{- 36 \pm \sqrt{1552 - 64 h}}{2 \cdot - 16}$

Factor $16$ out of $1552 - 64 h$ .

Factor $16$ out of $1552$ .

$t = \frac{- 36 \pm \sqrt{16 (97) - 64 h}}{2 \cdot - 16}$

Factor $16$ out of $- 64 h$ .

$t = \frac{- 36 \pm \sqrt{16 (97) + 16 (- 4 h)}}{2 \cdot - 16}$

Factor $16$ out of $16 (97) + 16 (- 4 h)$ .

$t = \frac{- 36 \pm \sqrt{16 (97 - 4 h)}}{2 \cdot - 16}$

Rewrite $16 (97 - 4 h)$ as ${(2^{2})}^{2} (97 - 4 h)$ .

Rewrite $16$ as $4^{2}$ .

$t = \frac{- 36 \pm \sqrt{4^{2} (97 - 4 h)}}{2 \cdot - 16}$

Rewrite $4$ as $2^{2}$ .

$t = \frac{- 36 \pm \sqrt{{(2^{2})}^{2} (97 - 4 h)}}{2 \cdot - 16}$

Pull terms out from under the radical.

$t = \frac{- 36 \pm 2^{2} \sqrt{97 - 4 h}}{2 \cdot - 16}$

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

$t = \frac{- 36 \pm 4 \sqrt{97 - 4 h}}{2 \cdot - 16}$

Multiply $2$ by $- 16$ .

$t = \frac{- 36 \pm 4 \sqrt{97 - 4 h}}{- 32}$

Simplify $\frac{- 36 \pm 4 \sqrt{97 - 4 h}}{- 32}$ .

$t = \frac{9 \pm \sqrt{97 - 4 h}}{8}$

Step 6

The final answer is the combination of both solutions.

$t = \frac{9 + \sqrt{97 - 4 h}}{8}$

$t = \frac{9 - \sqrt{97 - 4 h}}{8}$

Solve for t h=-16t^2+36t+4

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