Precalculus Examples

$h = - 40 - 5 t - 5 t^{2}$

Step 1

Rewrite the equation as $- 40 - 5 t - 5 t^{2} = h$ .

$- 40 - 5 t - 5 t^{2} = h$

Step 2

Subtract $h$ from both sides of the equation.

$- 40 - 5 t - 5 t^{2} - h = 0$

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 $a = - 5$ , $b = - 5$ , and $c = - 40 - h$ into the quadratic formula and solve for $t$ .

$\frac{5 \pm \sqrt{{(- 5)}^{2} - 4 \cdot (- 5 \cdot (- 40 - h))}}{2 \cdot - 5}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Factor $- 5$ out of ${(- 5)}^{2} - 4 \cdot - 5 \cdot (- 40 - h)$ .

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Step 5.1.1.1

Factor $- 5$ out of ${(- 5)}^{2}$ .

$t = \frac{5 \pm \sqrt{- 5 \cdot - 5 - 4 \cdot - 5 \cdot (- 40 - h)}}{2 \cdot - 5}$

Step 5.1.1.2

Factor $- 5$ out of $- 4 \cdot - 5 \cdot (- 40 - h)$ .

$t = \frac{5 \pm \sqrt{- 5 \cdot - 5 - 5 (- 4 \cdot (- 40 - h))}}{2 \cdot - 5}$

Step 5.1.1.3

Factor $- 5$ out of $- 5 \cdot - 5 - 5 (- 4 \cdot (- 40 - h))$ .

$t = \frac{5 \pm \sqrt{- 5 (- 5 - 4 \cdot (- 40 - h))}}{2 \cdot - 5}$

Step 5.1.2

Factor $- 1$ out of $- 5 - 4 \cdot (- 40 - h)$ .

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Step 5.1.2.1

Reorder the expression.

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Step 5.1.2.1.1

Reorder $- 40$ and $- h$ .

$t = \frac{5 \pm \sqrt{- 5 (- 5 - 4 \cdot (- h - 40))}}{2 \cdot - 5}$

Step 5.1.2.1.2

Reorder $- 5$ and $- 4 \cdot (- h - 40)$ .

$t = \frac{5 \pm \sqrt{- 5 (- 4 \cdot (- h - 40) - 5)}}{2 \cdot - 5}$

Step 5.1.2.2

Factor $- 1$ out of $- 4 \cdot (- h - 40)$ .

$t = \frac{5 \pm \sqrt{- 5 (- (4 \cdot (- h - 40)) - 5)}}{2 \cdot - 5}$

Step 5.1.2.3

Rewrite $- 5$ as $- 1 (5)$ .

$t = \frac{5 \pm \sqrt{- 5 (- (4 \cdot (- h - 40)) - 1 \cdot 5)}}{2 \cdot - 5}$

Step 5.1.2.4

Factor $- 1$ out of $- (4 \cdot (- h - 40)) - 1 (5)$ .

$t = \frac{5 \pm \sqrt{- 5 (- (4 \cdot (- h - 40) + 5))}}{2 \cdot - 5}$

Step 5.1.3

Apply the distributive property.

$t = \frac{5 \pm \sqrt{- 5 (- (4 (- h) + 4 \cdot - 40 + 5))}}{2 \cdot - 5}$

Step 5.1.4

Multiply $- 1$ by $4$ .

$t = \frac{5 \pm \sqrt{- 5 (- (- 4 h + 4 \cdot - 40 + 5))}}{2 \cdot - 5}$

Step 5.1.5

Multiply $4$ by $- 40$ .

$t = \frac{5 \pm \sqrt{- 5 (- (- 4 h - 160 + 5))}}{2 \cdot - 5}$

Step 5.1.6

Add $- 160$ and $5$ .

$t = \frac{5 \pm \sqrt{- 5 \cdot (- 1 (- 4 h - 155))}}{2 \cdot - 5}$

Step 5.1.7

Combine exponents.

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Step 5.1.7.1

Factor out negative.

$t = \frac{5 \pm \sqrt{- (- 5 (- 4 h - 155))}}{2 \cdot - 5}$

Step 5.1.7.2

Multiply $- 5$ by $- 1$ .

$t = \frac{5 \pm \sqrt{5 (- 4 h - 155)}}{2 \cdot - 5}$

Step 5.2

Multiply $2$ by $- 5$ .

$t = \frac{5 \pm \sqrt{5 (- 4 h - 155)}}{- 10}$

Step 5.3

Move the negative in front of the fraction.

$t = - \frac{5 \pm \sqrt{5 (- 4 h - 155)}}{10}$

Step 6

The final answer is the combination of both solutions.

$t = - \frac{5 + \sqrt{5 (- 4 h - 155)}}{10}$

$t = - \frac{5 - \sqrt{5 (- 4 h - 155)}}{10}$