Write as a Function of h h=-16t^2+36t+9

$h = - 16 t^{2} + 36 t + 9$

Rewrite the equation as $- 16 t^{2} + 36 t + 9 = h$ .

$- 16 t^{2} + 36 t + 9 = h$

Subtract $h$ from both sides of the equation.

$- 16 t^{2} + 36 t + 9 - h = 0$

Use the quadratic formula to find the solutions.

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

Substitute the values $a = - 16$ , $b = 36$ , and $c = 9 - h$ into the quadratic formula and solve for $t$ .

$\frac{- 36 \pm \sqrt{36^{2} - 4 \cdot (- 16 \cdot (9 - h))}}{2 \cdot - 16}$

Simplify.

Simplify the numerator.

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

$t = \frac{- 36 \pm \sqrt{1296 - 4 \cdot - 16 \cdot (9 - h)}}{2 \cdot - 16}$

Multiply $- 4$ by $- 16$ .

$t = \frac{- 36 \pm \sqrt{1296 + 64 \cdot (9 - h)}}{2 \cdot - 16}$

Apply the distributive property.

$t = \frac{- 36 \pm \sqrt{1296 + 64 \cdot 9 + 64 (- h)}}{2 \cdot - 16}$

Multiply $64$ by $9$ .

$t = \frac{- 36 \pm \sqrt{1296 + 576 + 64 (- h)}}{2 \cdot - 16}$

Multiply $- 1$ by $64$ .

$t = \frac{- 36 \pm \sqrt{1296 + 576 - 64 h}}{2 \cdot - 16}$

Add $1296$ and $576$ .

$t = \frac{- 36 \pm \sqrt{1872 - 64 h}}{2 \cdot - 16}$

Factor $16$ out of $1872 - 64 h$ .

Factor $16$ out of $1872$ .

$t = \frac{- 36 \pm \sqrt{16 (117) - 64 h}}{2 \cdot - 16}$

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

$t = \frac{- 36 \pm \sqrt{16 (117) + 16 (- 4 h)}}{2 \cdot - 16}$

Factor $16$ out of $16 (117) + 16 (- 4 h)$ .

$t = \frac{- 36 \pm \sqrt{16 (117 - 4 h)}}{2 \cdot - 16}$

Rewrite $16 (117 - 4 h)$ as ${(2^{2})}^{2} (117 - 4 h)$ .

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

$t = \frac{- 36 \pm \sqrt{4^{2} (117 - 4 h)}}{2 \cdot - 16}$

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

$t = \frac{- 36 \pm \sqrt{{(2^{2})}^{2} (117 - 4 h)}}{2 \cdot - 16}$

Pull terms out from under the radical.

$t = \frac{- 36 \pm 2^{2} \sqrt{117 - 4 h}}{2 \cdot - 16}$

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

$t = \frac{- 36 \pm 4 \sqrt{117 - 4 h}}{2 \cdot - 16}$

Multiply $2$ by $- 16$ .

$t = \frac{- 36 \pm 4 \sqrt{117 - 4 h}}{- 32}$

Simplify $\frac{- 36 \pm 4 \sqrt{117 - 4 h}}{- 32}$ .

$t = \frac{9 \pm \sqrt{117 - 4 h}}{8}$

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Simplify the numerator.

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

$t = \frac{- 36 \pm \sqrt{1296 - 4 \cdot - 16 \cdot (9 - h)}}{2 \cdot - 16}$

Multiply $- 4$ by $- 16$ .

$t = \frac{- 36 \pm \sqrt{1296 + 64 \cdot (9 - h)}}{2 \cdot - 16}$

Apply the distributive property.

$t = \frac{- 36 \pm \sqrt{1296 + 64 \cdot 9 + 64 (- h)}}{2 \cdot - 16}$

Multiply $64$ by $9$ .

$t = \frac{- 36 \pm \sqrt{1296 + 576 + 64 (- h)}}{2 \cdot - 16}$

Multiply $- 1$ by $64$ .

$t = \frac{- 36 \pm \sqrt{1296 + 576 - 64 h}}{2 \cdot - 16}$

Add $1296$ and $576$ .

$t = \frac{- 36 \pm \sqrt{1872 - 64 h}}{2 \cdot - 16}$

Factor $16$ out of $1872 - 64 h$ .

Factor $16$ out of $1872$ .

$t = \frac{- 36 \pm \sqrt{16 (117) - 64 h}}{2 \cdot - 16}$

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

$t = \frac{- 36 \pm \sqrt{16 (117) + 16 (- 4 h)}}{2 \cdot - 16}$

Factor $16$ out of $16 (117) + 16 (- 4 h)$ .

$t = \frac{- 36 \pm \sqrt{16 (117 - 4 h)}}{2 \cdot - 16}$

Rewrite $16 (117 - 4 h)$ as ${(2^{2})}^{2} (117 - 4 h)$ .

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

$t = \frac{- 36 \pm \sqrt{4^{2} (117 - 4 h)}}{2 \cdot - 16}$

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

$t = \frac{- 36 \pm \sqrt{{(2^{2})}^{2} (117 - 4 h)}}{2 \cdot - 16}$

Pull terms out from under the radical.

$t = \frac{- 36 \pm 2^{2} \sqrt{117 - 4 h}}{2 \cdot - 16}$

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

$t = \frac{- 36 \pm 4 \sqrt{117 - 4 h}}{2 \cdot - 16}$

Multiply $2$ by $- 16$ .

$t = \frac{- 36 \pm 4 \sqrt{117 - 4 h}}{- 32}$

Simplify $\frac{- 36 \pm 4 \sqrt{117 - 4 h}}{- 32}$ .

$t = \frac{9 \pm \sqrt{117 - 4 h}}{8}$

Change the $\pm$ to $+$ .

$t = \frac{9 + \sqrt{117 - 4 h}}{8}$

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Simplify the numerator.

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

$t = \frac{- 36 \pm \sqrt{1296 - 4 \cdot - 16 \cdot (9 - h)}}{2 \cdot - 16}$

Multiply $- 4$ by $- 16$ .

$t = \frac{- 36 \pm \sqrt{1296 + 64 \cdot (9 - h)}}{2 \cdot - 16}$

Apply the distributive property.

$t = \frac{- 36 \pm \sqrt{1296 + 64 \cdot 9 + 64 (- h)}}{2 \cdot - 16}$

Multiply $64$ by $9$ .

$t = \frac{- 36 \pm \sqrt{1296 + 576 + 64 (- h)}}{2 \cdot - 16}$

Multiply $- 1$ by $64$ .

$t = \frac{- 36 \pm \sqrt{1296 + 576 - 64 h}}{2 \cdot - 16}$

Add $1296$ and $576$ .

$t = \frac{- 36 \pm \sqrt{1872 - 64 h}}{2 \cdot - 16}$

Factor $16$ out of $1872 - 64 h$ .

Factor $16$ out of $1872$ .

$t = \frac{- 36 \pm \sqrt{16 (117) - 64 h}}{2 \cdot - 16}$

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

$t = \frac{- 36 \pm \sqrt{16 (117) + 16 (- 4 h)}}{2 \cdot - 16}$

Factor $16$ out of $16 (117) + 16 (- 4 h)$ .

$t = \frac{- 36 \pm \sqrt{16 (117 - 4 h)}}{2 \cdot - 16}$

Rewrite $16 (117 - 4 h)$ as ${(2^{2})}^{2} (117 - 4 h)$ .

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

$t = \frac{- 36 \pm \sqrt{4^{2} (117 - 4 h)}}{2 \cdot - 16}$

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

$t = \frac{- 36 \pm \sqrt{{(2^{2})}^{2} (117 - 4 h)}}{2 \cdot - 16}$

Pull terms out from under the radical.

$t = \frac{- 36 \pm 2^{2} \sqrt{117 - 4 h}}{2 \cdot - 16}$

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

$t = \frac{- 36 \pm 4 \sqrt{117 - 4 h}}{2 \cdot - 16}$

Multiply $2$ by $- 16$ .

$t = \frac{- 36 \pm 4 \sqrt{117 - 4 h}}{- 32}$

Simplify $\frac{- 36 \pm 4 \sqrt{117 - 4 h}}{- 32}$ .

$t = \frac{9 \pm \sqrt{117 - 4 h}}{8}$

Change the $\pm$ to $-$ .

$t = \frac{9 - \sqrt{117 - 4 h}}{8}$

The final answer is the combination of both solutions.

$t = \frac{9 + \sqrt{117 - 4 h}}{8}$

$t = \frac{9 - \sqrt{117 - 4 h}}{8}$

Write as a Function of h h=-16t^2+36t+9

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