Write as a Function of h h=-5t^2+30t+10

$h = - 5 t^{2} + 30 t + 10$

Rewrite the equation as $- 5 t^{2} + 30 t + 10 = h$ .

$- 5 t^{2} + 30 t + 10 = h$

Subtract $h$ from both sides of the equation.

$- 5 t^{2} + 30 t + 10 - h = 0$

Use the quadratic formula to find the solutions.

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

Substitute the values $a = - 5$ , $b = 30$ , and $c = 10 - h$ into the quadratic formula and solve for $t$ .

$\frac{- 30 \pm \sqrt{30^{2} - 4 \cdot (- 5 \cdot (10 - h))}}{2 \cdot - 5}$

Simplify.

Simplify the numerator.

Raise $30$ to the power of $2$ .

$t = \frac{- 30 \pm \sqrt{900 - 4 \cdot - 5 \cdot (10 - h)}}{2 \cdot - 5}$

Multiply $- 4$ by $- 5$ .

$t = \frac{- 30 \pm \sqrt{900 + 20 \cdot (10 - h)}}{2 \cdot - 5}$

Apply the distributive property.

$t = \frac{- 30 \pm \sqrt{900 + 20 \cdot 10 + 20 (- h)}}{2 \cdot - 5}$

Multiply $20$ by $10$ .

$t = \frac{- 30 \pm \sqrt{900 + 200 + 20 (- h)}}{2 \cdot - 5}$

Multiply $- 1$ by $20$ .

$t = \frac{- 30 \pm \sqrt{900 + 200 - 20 h}}{2 \cdot - 5}$

Add $900$ and $200$ .

$t = \frac{- 30 \pm \sqrt{1100 - 20 h}}{2 \cdot - 5}$

Factor $20$ out of $1100 - 20 h$ .

Factor $20$ out of $1100$ .

$t = \frac{- 30 \pm \sqrt{20 (55) - 20 h}}{2 \cdot - 5}$

Factor $20$ out of $- 20 h$ .

$t = \frac{- 30 \pm \sqrt{20 (55) + 20 (- h)}}{2 \cdot - 5}$

Factor $20$ out of $20 (55) + 20 (- h)$ .

$t = \frac{- 30 \pm \sqrt{20 (55 - h)}}{2 \cdot - 5}$

Rewrite $20 (55 - h)$ as $2^{2} \cdot (5 (55 - h))$ .

Factor $4$ out of $20$ .

$t = \frac{- 30 \pm \sqrt{4 (5) (55 - h)}}{2 \cdot - 5}$

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

$t = \frac{- 30 \pm \sqrt{2^{2} \cdot (5 (55 - h))}}{2 \cdot - 5}$

Add parentheses.

$t = \frac{- 30 \pm \sqrt{2^{2} \cdot (5 (55 - h))}}{2 \cdot - 5}$

Pull terms out from under the radical.

$t = \frac{- 30 \pm 2 \sqrt{5 (55 - h)}}{2 \cdot - 5}$

Multiply $2$ by $- 5$ .

$t = \frac{- 30 \pm 2 \sqrt{5 (55 - h)}}{- 10}$

Simplify $\frac{- 30 \pm 2 \sqrt{5 (55 - h)}}{- 10}$ .

$t = \frac{15 \pm \sqrt{5 (55 - h)}}{5}$

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

Simplify the numerator.

Raise $30$ to the power of $2$ .

$t = \frac{- 30 \pm \sqrt{900 - 4 \cdot - 5 \cdot (10 - h)}}{2 \cdot - 5}$

Multiply $- 4$ by $- 5$ .

$t = \frac{- 30 \pm \sqrt{900 + 20 \cdot (10 - h)}}{2 \cdot - 5}$

Apply the distributive property.

$t = \frac{- 30 \pm \sqrt{900 + 20 \cdot 10 + 20 (- h)}}{2 \cdot - 5}$

Multiply $20$ by $10$ .

$t = \frac{- 30 \pm \sqrt{900 + 200 + 20 (- h)}}{2 \cdot - 5}$

Multiply $- 1$ by $20$ .

$t = \frac{- 30 \pm \sqrt{900 + 200 - 20 h}}{2 \cdot - 5}$

Add $900$ and $200$ .

$t = \frac{- 30 \pm \sqrt{1100 - 20 h}}{2 \cdot - 5}$

Factor $20$ out of $1100 - 20 h$ .

Factor $20$ out of $1100$ .

$t = \frac{- 30 \pm \sqrt{20 (55) - 20 h}}{2 \cdot - 5}$

Factor $20$ out of $- 20 h$ .

$t = \frac{- 30 \pm \sqrt{20 (55) + 20 (- h)}}{2 \cdot - 5}$

Factor $20$ out of $20 (55) + 20 (- h)$ .

$t = \frac{- 30 \pm \sqrt{20 (55 - h)}}{2 \cdot - 5}$

Rewrite $20 (55 - h)$ as $2^{2} \cdot (5 (55 - h))$ .

Factor $4$ out of $20$ .

$t = \frac{- 30 \pm \sqrt{4 (5) (55 - h)}}{2 \cdot - 5}$

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

$t = \frac{- 30 \pm \sqrt{2^{2} \cdot (5 (55 - h))}}{2 \cdot - 5}$

Add parentheses.

$t = \frac{- 30 \pm \sqrt{2^{2} \cdot (5 (55 - h))}}{2 \cdot - 5}$

Pull terms out from under the radical.

$t = \frac{- 30 \pm 2 \sqrt{5 (55 - h)}}{2 \cdot - 5}$

Multiply $2$ by $- 5$ .

$t = \frac{- 30 \pm 2 \sqrt{5 (55 - h)}}{- 10}$

Simplify $\frac{- 30 \pm 2 \sqrt{5 (55 - h)}}{- 10}$ .

$t = \frac{15 \pm \sqrt{5 (55 - h)}}{5}$

Change the $\pm$ to $+$ .

$t = \frac{15 + \sqrt{5 (55 - h)}}{5}$

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

Simplify the numerator.

Raise $30$ to the power of $2$ .

$t = \frac{- 30 \pm \sqrt{900 - 4 \cdot - 5 \cdot (10 - h)}}{2 \cdot - 5}$

Multiply $- 4$ by $- 5$ .

$t = \frac{- 30 \pm \sqrt{900 + 20 \cdot (10 - h)}}{2 \cdot - 5}$

Apply the distributive property.

$t = \frac{- 30 \pm \sqrt{900 + 20 \cdot 10 + 20 (- h)}}{2 \cdot - 5}$

Multiply $20$ by $10$ .

$t = \frac{- 30 \pm \sqrt{900 + 200 + 20 (- h)}}{2 \cdot - 5}$

Multiply $- 1$ by $20$ .

$t = \frac{- 30 \pm \sqrt{900 + 200 - 20 h}}{2 \cdot - 5}$

Add $900$ and $200$ .

$t = \frac{- 30 \pm \sqrt{1100 - 20 h}}{2 \cdot - 5}$

Factor $20$ out of $1100 - 20 h$ .

Factor $20$ out of $1100$ .

$t = \frac{- 30 \pm \sqrt{20 (55) - 20 h}}{2 \cdot - 5}$

Factor $20$ out of $- 20 h$ .

$t = \frac{- 30 \pm \sqrt{20 (55) + 20 (- h)}}{2 \cdot - 5}$

Factor $20$ out of $20 (55) + 20 (- h)$ .

$t = \frac{- 30 \pm \sqrt{20 (55 - h)}}{2 \cdot - 5}$

Rewrite $20 (55 - h)$ as $2^{2} \cdot (5 (55 - h))$ .

Factor $4$ out of $20$ .

$t = \frac{- 30 \pm \sqrt{4 (5) (55 - h)}}{2 \cdot - 5}$

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

$t = \frac{- 30 \pm \sqrt{2^{2} \cdot (5 (55 - h))}}{2 \cdot - 5}$

Add parentheses.

$t = \frac{- 30 \pm \sqrt{2^{2} \cdot (5 (55 - h))}}{2 \cdot - 5}$

Pull terms out from under the radical.

$t = \frac{- 30 \pm 2 \sqrt{5 (55 - h)}}{2 \cdot - 5}$

Multiply $2$ by $- 5$ .

$t = \frac{- 30 \pm 2 \sqrt{5 (55 - h)}}{- 10}$

Simplify $\frac{- 30 \pm 2 \sqrt{5 (55 - h)}}{- 10}$ .

$t = \frac{15 \pm \sqrt{5 (55 - h)}}{5}$

Change the $\pm$ to $-$ .

$t = \frac{15 - \sqrt{5 (55 - h)}}{5}$

The final answer is the combination of both solutions.

$t = \frac{15 + \sqrt{5 (55 - h)}}{5}$

$t = \frac{15 - \sqrt{5 (55 - h)}}{5}$

Write as a Function of h h=-5t^2+30t+10

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