Precalculus Examples

$80 (t) = 96 + 43 t - 16 t^{2}$

Step 1

Since $t$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$96 + 43 t - 16 t^{2} = 80 t$

Step 2

Move all terms containing $t$ to the left side of the equation.

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Step 2.1

Subtract $80 t$ from both sides of the equation.

$96 + 43 t - 16 t^{2} - 80 t = 0$

Step 2.2

Subtract $80 t$ from $43 t$ .

$96 - 16 t^{2} - 37 t = 0$

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 $a = - 16$ , $b = - 37$ , and $c = 96$ into the quadratic formula and solve for $t$ .

$\frac{37 \pm \sqrt{{(- 37)}^{2} - 4 \cdot (- 16 \cdot 96)}}{2 \cdot - 16}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $- 37$ to the power of $2$ .

$t = \frac{37 \pm \sqrt{1369 - 4 \cdot - 16 \cdot 96}}{2 \cdot - 16}$

Step 5.1.2

Multiply $- 4 \cdot - 16 \cdot 96$ .

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Step 5.1.2.1

Multiply $- 4$ by $- 16$ .

$t = \frac{37 \pm \sqrt{1369 + 64 \cdot 96}}{2 \cdot - 16}$

Step 5.1.2.2

Multiply $64$ by $96$ .

$t = \frac{37 \pm \sqrt{1369 + 6144}}{2 \cdot - 16}$

Step 5.1.3

Add $1369$ and $6144$ .

$t = \frac{37 \pm \sqrt{7513}}{2 \cdot - 16}$

Step 5.2

Multiply $2$ by $- 16$ .

$t = \frac{37 \pm \sqrt{7513}}{- 32}$

Step 5.3

Move the negative in front of the fraction.

$t = - \frac{37 \pm \sqrt{7513}}{32}$

Step 6

The final answer is the combination of both solutions.

$t = - \frac{37 + \sqrt{7513}}{32}, - \frac{37 - \sqrt{7513}}{32}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$t = - \frac{37 + \sqrt{7513}}{32}, - \frac{37 - \sqrt{7513}}{32}$

Decimal Form:

$t = - 3.86492385 \dots, 1.55242385 \dots$