Write as a Function of h h=-16t^2+25t+6

$h = - 16 t^{2} + 25 t + 6$

Step 1

Rewrite the equation as $- 16 t^{2} + 25 t + 6 = h$ .

$- 16 t^{2} + 25 t + 6 = h$

Step 2

Subtract $h$ from both sides of the equation.

$- 16 t^{2} + 25 t + 6 - h = 0$

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 $a = - 16$ , $b = 25$ , and $c = 6 - h$ into the quadratic formula and solve for $t$ .

$\frac{- 25 \pm \sqrt{25^{2} - 4 \cdot (- 16 \cdot (6 - h))}}{2 \cdot - 16}$

Step 5

Simplify.

Simplify the numerator.

Raise $25$ to the power of $2$ .

$t = \frac{- 25 \pm \sqrt{625 - 4 \cdot - 16 \cdot (6 - h)}}{2 \cdot - 16}$

Multiply $- 4$ by $- 16$ .

$t = \frac{- 25 \pm \sqrt{625 + 64 \cdot (6 - h)}}{2 \cdot - 16}$

Apply the distributive property.

$t = \frac{- 25 \pm \sqrt{625 + 64 \cdot 6 + 64 (- h)}}{2 \cdot - 16}$

Multiply $64$ by $6$ .

$t = \frac{- 25 \pm \sqrt{625 + 384 + 64 (- h)}}{2 \cdot - 16}$

Multiply $- 1$ by $64$ .

$t = \frac{- 25 \pm \sqrt{625 + 384 - 64 h}}{2 \cdot - 16}$

Add $625$ and $384$ .

$t = \frac{- 25 \pm \sqrt{1009 - 64 h}}{2 \cdot - 16}$

Multiply $2$ by $- 16$ .

$t = \frac{- 25 \pm \sqrt{1009 - 64 h}}{- 32}$

Simplify $\frac{- 25 \pm \sqrt{1009 - 64 h}}{- 32}$ .

$t = \frac{25 \pm \sqrt{1009 - 64 h}}{32}$

Step 6

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

Simplify the numerator.

Raise $25$ to the power of $2$ .

$t = \frac{- 25 \pm \sqrt{625 - 4 \cdot - 16 \cdot (6 - h)}}{2 \cdot - 16}$

Multiply $- 4$ by $- 16$ .

$t = \frac{- 25 \pm \sqrt{625 + 64 \cdot (6 - h)}}{2 \cdot - 16}$

Apply the distributive property.

$t = \frac{- 25 \pm \sqrt{625 + 64 \cdot 6 + 64 (- h)}}{2 \cdot - 16}$

Multiply $64$ by $6$ .

$t = \frac{- 25 \pm \sqrt{625 + 384 + 64 (- h)}}{2 \cdot - 16}$

Multiply $- 1$ by $64$ .

$t = \frac{- 25 \pm \sqrt{625 + 384 - 64 h}}{2 \cdot - 16}$

Add $625$ and $384$ .

$t = \frac{- 25 \pm \sqrt{1009 - 64 h}}{2 \cdot - 16}$

Multiply $2$ by $- 16$ .

$t = \frac{- 25 \pm \sqrt{1009 - 64 h}}{- 32}$

Simplify $\frac{- 25 \pm \sqrt{1009 - 64 h}}{- 32}$ .

$t = \frac{25 \pm \sqrt{1009 - 64 h}}{32}$

Change the $\pm$ to $+$ .

$t = \frac{25 + \sqrt{1009 - 64 h}}{32}$

Step 7

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

Simplify the numerator.

Raise $25$ to the power of $2$ .

$t = \frac{- 25 \pm \sqrt{625 - 4 \cdot - 16 \cdot (6 - h)}}{2 \cdot - 16}$

Multiply $- 4$ by $- 16$ .

$t = \frac{- 25 \pm \sqrt{625 + 64 \cdot (6 - h)}}{2 \cdot - 16}$

Apply the distributive property.

$t = \frac{- 25 \pm \sqrt{625 + 64 \cdot 6 + 64 (- h)}}{2 \cdot - 16}$

Multiply $64$ by $6$ .

$t = \frac{- 25 \pm \sqrt{625 + 384 + 64 (- h)}}{2 \cdot - 16}$

Multiply $- 1$ by $64$ .

$t = \frac{- 25 \pm \sqrt{625 + 384 - 64 h}}{2 \cdot - 16}$

Add $625$ and $384$ .

$t = \frac{- 25 \pm \sqrt{1009 - 64 h}}{2 \cdot - 16}$

Multiply $2$ by $- 16$ .

$t = \frac{- 25 \pm \sqrt{1009 - 64 h}}{- 32}$

Simplify $\frac{- 25 \pm \sqrt{1009 - 64 h}}{- 32}$ .

$t = \frac{25 \pm \sqrt{1009 - 64 h}}{32}$

Change the $\pm$ to $-$ .

$t = \frac{25 - \sqrt{1009 - 64 h}}{32}$

Step 8

The final answer is the combination of both solutions.

$t = \frac{25 + \sqrt{1009 - 64 h}}{32}$

$t = \frac{25 - \sqrt{1009 - 64 h}}{32}$

Write as a Function of h h=-16t^2+25t+6

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