Write as a Function of h h=-16t^2+32t+6

$h = - 16 t^{2} + 32 t + 6$

Step 1

Rewrite the equation as $- 16 t^{2} + 32 t + 6 = h$ .

$- 16 t^{2} + 32 t + 6 = h$

Step 2

Subtract $h$ from both sides of the equation.

$- 16 t^{2} + 32 t + 6 - h = 0$

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 $a = - 16$ , $b = 32$ , and $c = 6 - h$ into the quadratic formula and solve for $t$ .

$\frac{- 32 \pm \sqrt{32^{2} - 4 \cdot (- 16 \cdot (6 - h))}}{2 \cdot - 16}$

Step 5

Simplify.

Simplify the numerator.

Raise $32$ to the power of $2$ .

$t = \frac{- 32 \pm \sqrt{1024 - 4 \cdot - 16 \cdot (6 - h)}}{2 \cdot - 16}$

Multiply $- 4$ by $- 16$ .

$t = \frac{- 32 \pm \sqrt{1024 + 64 \cdot (6 - h)}}{2 \cdot - 16}$

Apply the distributive property.

$t = \frac{- 32 \pm \sqrt{1024 + 64 \cdot 6 + 64 (- h)}}{2 \cdot - 16}$

Multiply $64$ by $6$ .

$t = \frac{- 32 \pm \sqrt{1024 + 384 + 64 (- h)}}{2 \cdot - 16}$

Multiply $- 1$ by $64$ .

$t = \frac{- 32 \pm \sqrt{1024 + 384 - 64 h}}{2 \cdot - 16}$

Add $1024$ and $384$ .

$t = \frac{- 32 \pm \sqrt{1408 - 64 h}}{2 \cdot - 16}$

Factor $64$ out of $1408 - 64 h$ .

Factor $64$ out of $1408$ .

$t = \frac{- 32 \pm \sqrt{64 (22) - 64 h}}{2 \cdot - 16}$

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

$t = \frac{- 32 \pm \sqrt{64 (22) + 64 (- h)}}{2 \cdot - 16}$

Factor $64$ out of $64 (22) + 64 (- h)$ .

$t = \frac{- 32 \pm \sqrt{64 (22 - h)}}{2 \cdot - 16}$

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

$t = \frac{- 32 \pm \sqrt{8^{2} (22 - h)}}{2 \cdot - 16}$

Pull terms out from under the radical.

$t = \frac{- 32 \pm 8 \sqrt{22 - h}}{2 \cdot - 16}$

Multiply $2$ by $- 16$ .

$t = \frac{- 32 \pm 8 \sqrt{22 - h}}{- 32}$

Simplify $\frac{- 32 \pm 8 \sqrt{22 - h}}{- 32}$ .

$t = \frac{4 \pm \sqrt{22 - h}}{4}$

Step 6

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

Simplify the numerator.

Raise $32$ to the power of $2$ .

$t = \frac{- 32 \pm \sqrt{1024 - 4 \cdot - 16 \cdot (6 - h)}}{2 \cdot - 16}$

Multiply $- 4$ by $- 16$ .

$t = \frac{- 32 \pm \sqrt{1024 + 64 \cdot (6 - h)}}{2 \cdot - 16}$

Apply the distributive property.

$t = \frac{- 32 \pm \sqrt{1024 + 64 \cdot 6 + 64 (- h)}}{2 \cdot - 16}$

Multiply $64$ by $6$ .

$t = \frac{- 32 \pm \sqrt{1024 + 384 + 64 (- h)}}{2 \cdot - 16}$

Multiply $- 1$ by $64$ .

$t = \frac{- 32 \pm \sqrt{1024 + 384 - 64 h}}{2 \cdot - 16}$

Add $1024$ and $384$ .

$t = \frac{- 32 \pm \sqrt{1408 - 64 h}}{2 \cdot - 16}$

Factor $64$ out of $1408 - 64 h$ .

Factor $64$ out of $1408$ .

$t = \frac{- 32 \pm \sqrt{64 (22) - 64 h}}{2 \cdot - 16}$

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

$t = \frac{- 32 \pm \sqrt{64 (22) + 64 (- h)}}{2 \cdot - 16}$

Factor $64$ out of $64 (22) + 64 (- h)$ .

$t = \frac{- 32 \pm \sqrt{64 (22 - h)}}{2 \cdot - 16}$

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

$t = \frac{- 32 \pm \sqrt{8^{2} (22 - h)}}{2 \cdot - 16}$

Pull terms out from under the radical.

$t = \frac{- 32 \pm 8 \sqrt{22 - h}}{2 \cdot - 16}$

Multiply $2$ by $- 16$ .

$t = \frac{- 32 \pm 8 \sqrt{22 - h}}{- 32}$

Simplify $\frac{- 32 \pm 8 \sqrt{22 - h}}{- 32}$ .

$t = \frac{4 \pm \sqrt{22 - h}}{4}$

Change the $\pm$ to $+$ .

$t = \frac{4 + \sqrt{22 - h}}{4}$

Step 7

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

Simplify the numerator.

Raise $32$ to the power of $2$ .

$t = \frac{- 32 \pm \sqrt{1024 - 4 \cdot - 16 \cdot (6 - h)}}{2 \cdot - 16}$

Multiply $- 4$ by $- 16$ .

$t = \frac{- 32 \pm \sqrt{1024 + 64 \cdot (6 - h)}}{2 \cdot - 16}$

Apply the distributive property.

$t = \frac{- 32 \pm \sqrt{1024 + 64 \cdot 6 + 64 (- h)}}{2 \cdot - 16}$

Multiply $64$ by $6$ .

$t = \frac{- 32 \pm \sqrt{1024 + 384 + 64 (- h)}}{2 \cdot - 16}$

Multiply $- 1$ by $64$ .

$t = \frac{- 32 \pm \sqrt{1024 + 384 - 64 h}}{2 \cdot - 16}$

Add $1024$ and $384$ .

$t = \frac{- 32 \pm \sqrt{1408 - 64 h}}{2 \cdot - 16}$

Factor $64$ out of $1408 - 64 h$ .

Factor $64$ out of $1408$ .

$t = \frac{- 32 \pm \sqrt{64 (22) - 64 h}}{2 \cdot - 16}$

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

$t = \frac{- 32 \pm \sqrt{64 (22) + 64 (- h)}}{2 \cdot - 16}$

Factor $64$ out of $64 (22) + 64 (- h)$ .

$t = \frac{- 32 \pm \sqrt{64 (22 - h)}}{2 \cdot - 16}$

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

$t = \frac{- 32 \pm \sqrt{8^{2} (22 - h)}}{2 \cdot - 16}$

Pull terms out from under the radical.

$t = \frac{- 32 \pm 8 \sqrt{22 - h}}{2 \cdot - 16}$

Multiply $2$ by $- 16$ .

$t = \frac{- 32 \pm 8 \sqrt{22 - h}}{- 32}$

Simplify $\frac{- 32 \pm 8 \sqrt{22 - h}}{- 32}$ .

$t = \frac{4 \pm \sqrt{22 - h}}{4}$

Change the $\pm$ to $-$ .

$t = \frac{4 - \sqrt{22 - h}}{4}$

Step 8

The final answer is the combination of both solutions.

$t = \frac{4 + \sqrt{22 - h}}{4}$

$t = \frac{4 - \sqrt{22 - h}}{4}$

Write as a Function of h h=-16t^2+32t+6

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