Trigonometry Examples

$h = 32 - 2 t - 5 t^{2}$

Step 1

Rewrite the equation as $32 - 2 t - 5 t^{2} = h$ .

$32 - 2 t - 5 t^{2} = h$

Step 2

Subtract $h$ from both sides of the equation.

$32 - 2 t - 5 t^{2} - h = 0$

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 $a = - 5$ , $b = - 2$ , and $c = 32 - h$ into the quadratic formula and solve for $t$ .

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (- 5 \cdot (32 - h))}}{2 \cdot - 5}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

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

$t = \frac{2 \pm \sqrt{4 - 4 \cdot - 5 \cdot (32 - h)}}{2 \cdot - 5}$

Step 5.1.2

Multiply $- 4$ by $- 5$ .

$t = \frac{2 \pm \sqrt{4 + 20 \cdot (32 - h)}}{2 \cdot - 5}$

Step 5.1.3

Apply the distributive property.

$t = \frac{2 \pm \sqrt{4 + 20 \cdot 32 + 20 (- h)}}{2 \cdot - 5}$

Step 5.1.4

Multiply $20$ by $32$ .

$t = \frac{2 \pm \sqrt{4 + 640 + 20 (- h)}}{2 \cdot - 5}$

Step 5.1.5

Multiply $- 1$ by $20$ .

$t = \frac{2 \pm \sqrt{4 + 640 - 20 h}}{2 \cdot - 5}$

Step 5.1.6

Add $4$ and $640$ .

$t = \frac{2 \pm \sqrt{644 - 20 h}}{2 \cdot - 5}$

Step 5.1.7

Factor $4$ out of $644 - 20 h$ .

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Step 5.1.7.1

Factor $4$ out of $644$ .

$t = \frac{2 \pm \sqrt{4 (161) - 20 h}}{2 \cdot - 5}$

Step 5.1.7.2

Factor $4$ out of $- 20 h$ .

$t = \frac{2 \pm \sqrt{4 (161) + 4 (- 5 h)}}{2 \cdot - 5}$

Step 5.1.7.3

Factor $4$ out of $4 (161) + 4 (- 5 h)$ .

$t = \frac{2 \pm \sqrt{4 (161 - 5 h)}}{2 \cdot - 5}$

Step 5.1.8

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

$t = \frac{2 \pm \sqrt{2^{2} (161 - 5 h)}}{2 \cdot - 5}$

Step 5.1.9

Pull terms out from under the radical.

$t = \frac{2 \pm 2 \sqrt{161 - 5 h}}{2 \cdot - 5}$

Step 5.2

Multiply $2$ by $- 5$ .

$t = \frac{2 \pm 2 \sqrt{161 - 5 h}}{- 10}$

Step 5.3

Simplify $\frac{2 \pm 2 \sqrt{161 - 5 h}}{- 10}$ .

$t = \frac{1 \pm \sqrt{161 - 5 h}}{- 5}$

Step 5.4

Move the negative in front of the fraction.

$t = - \frac{1 \pm \sqrt{161 - 5 h}}{5}$

Step 6

The final answer is the combination of both solutions.

$t = - \frac{1 + \sqrt{161 - 5 h}}{5}$

$t = - \frac{1 - \sqrt{161 - 5 h}}{5}$