Precalculus Examples

$h = - 9.8 t^{2} + 88.2 t$

Step 1

Rewrite the equation as $- 9.8 t^{2} + 88.2 t = h$ .

$- 9.8 t^{2} + 88.2 t = h$

Step 2

Subtract $h$ from both sides of the equation.

$- 9.8 t^{2} + 88.2 t - h = 0$

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 $a = - 9.8$ , $b = 88.2$ , and $c = - h$ into the quadratic formula and solve for $t$ .

$\frac{- 88.2 \pm \sqrt{{88.2}^{2} - 4 \cdot (- 9.8 \cdot (- h))}}{2 \cdot - 9.8}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $88.2$ to the power of $2$ .

$t = \frac{- 88.2 \pm \sqrt{7779.24 - 4 \cdot - 9.8 \cdot (- 1 h)}}{2 \cdot - 9.8}$

Step 5.1.2

Multiply $- 4 \cdot - 9.8 \cdot - 1$ .

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Step 5.1.2.1

Multiply $- 4$ by $- 9.8$ .

$t = \frac{- 88.2 \pm \sqrt{7779.24 + 39.2 \cdot (- 1 h)}}{2 \cdot - 9.8}$

Step 5.1.2.2

Multiply $39.2$ by $- 1$ .

$t = \frac{- 88.2 \pm \sqrt{7779.24 - 39.2 h}}{2 \cdot - 9.8}$

Step 5.1.3

Factor $1.96$ out of $7779.24 - 39.2 h$ .

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Step 5.1.3.1

Factor $1.96$ out of $7779.24$ .

$t = \frac{- 88.2 \pm \sqrt{1.96 (3969) - 39.2 h}}{2 \cdot - 9.8}$

Step 5.1.3.2

Factor $1.96$ out of $- 39.2 h$ .

$t = \frac{- 88.2 \pm \sqrt{1.96 (3969) + 1.96 (- 20 h)}}{2 \cdot - 9.8}$

Step 5.1.3.3

Factor $1.96$ out of $1.96 (3969) + 1.96 (- 20 h)$ .

$t = \frac{- 88.2 \pm \sqrt{1.96 (3969 - 20 h)}}{2 \cdot - 9.8}$

Step 5.1.4

Rewrite $1.96 (3969 - 20 h)$ as ${({1.18321595}^{2})}^{2} (3969 - 20 h)$ .

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Step 5.1.4.1

Rewrite $1.96$ as ${1.4}^{2}$ .

$t = \frac{- 88.2 \pm \sqrt{{1.4}^{2} (3969 - 20 h)}}{2 \cdot - 9.8}$

Step 5.1.4.2

Rewrite $1.4$ as ${1.18321595}^{2}$ .

$t = \frac{- 88.2 \pm \sqrt{{({1.18321595}^{2})}^{2} (3969 - 20 h)}}{2 \cdot - 9.8}$

Step 5.1.5

Pull terms out from under the radical.

$t = \frac{- 88.2 \pm {1.18321595}^{2} \sqrt{3969 - 20 h}}{2 \cdot - 9.8}$

Step 5.1.6

Raise $1.18321595$ to the power of $2$ .

$t = \frac{- 88.2 \pm 1.3\overline{9} \sqrt{3969 - 20 h}}{2 \cdot - 9.8}$

Step 5.2

Multiply $2$ by $- 9.8$ .

$t = \frac{- 88.2 \pm 1.3\overline{9} \sqrt{3969 - 20 h}}{- 19.6}$

Step 5.3

Simplify $\frac{- 88.2 \pm 1.3\overline{9} \sqrt{3969 - 20 h}}{- 19.6}$ .

$t = \frac{441 \pm 7 \sqrt{3969 - 20 h}}{98}$

Step 6

The final answer is the combination of both solutions.

$t = \frac{441 + 7 \sqrt{3969 - 20 h}}{98}$

$t = \frac{441 - 7 \sqrt{3969 - 20 h}}{98}$