Solve for t h=50-10t-5t^2

$h = 50 - 10 t - 5 t^{2}$

Step 1

Rewrite the equation as $50 - 10 t - 5 t^{2} = h$ .

$50 - 10 t - 5 t^{2} = h$

Step 2

Subtract $h$ from both sides of the equation.

$50 - 10 t - 5 t^{2} - h = 0$

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 $a = - 5$ , $b = - 10$ , and $c = 50 - h$ into the quadratic formula and solve for $t$ .

$\frac{10 \pm \sqrt{{(- 10)}^{2} - 4 \cdot (- 5 \cdot (50 - h))}}{2 \cdot - 5}$

Step 5

Simplify.

Simplify the numerator.

Raise $- 10$ to the power of $2$ .

$t = \frac{10 \pm \sqrt{100 - 4 \cdot - 5 \cdot (50 - h)}}{2 \cdot - 5}$

Multiply $- 4$ by $- 5$ .

$t = \frac{10 \pm \sqrt{100 + 20 \cdot (50 - h)}}{2 \cdot - 5}$

Apply the distributive property.

$t = \frac{10 \pm \sqrt{100 + 20 \cdot 50 + 20 (- h)}}{2 \cdot - 5}$

Multiply $20$ by $50$ .

$t = \frac{10 \pm \sqrt{100 + 1000 + 20 (- h)}}{2 \cdot - 5}$

Multiply $- 1$ by $20$ .

$t = \frac{10 \pm \sqrt{100 + 1000 - 20 h}}{2 \cdot - 5}$

Add $100$ and $1000$ .

$t = \frac{10 \pm \sqrt{1100 - 20 h}}{2 \cdot - 5}$

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

Factor $20$ out of $1100$ .

$t = \frac{10 \pm \sqrt{20 (55) - 20 h}}{2 \cdot - 5}$

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

$t = \frac{10 \pm \sqrt{20 (55) + 20 (- h)}}{2 \cdot - 5}$

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

$t = \frac{10 \pm \sqrt{20 (55 - h)}}{2 \cdot - 5}$

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

Factor $4$ out of $20$ .

$t = \frac{10 \pm \sqrt{4 (5) (55 - h)}}{2 \cdot - 5}$

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

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

Add parentheses.

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

Pull terms out from under the radical.

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

Multiply $2$ by $- 5$ .

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

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

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

Move the negative in front of the fraction.

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

Step 6

The final answer is the combination of both solutions.

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

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

Solve for t h=50-10t-5t^2

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