Precalculus Examples

$0 = (- 16) t^{2} + (60) t + 80$

Step 1

Move all the expressions to the left side of the equation.

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Step 1.1

Subtract $(- 16) t^{2}$ from both sides of the equation.

$16 t^{2} = (60) t + 80$

Step 1.2

Subtract $(60) t$ from both sides of the equation.

$16 t^{2} - 60 t = 80$

Step 1.3

Subtract $80$ from both sides of the equation.

$16 t^{2} - 60 t - 80 = 0$

Step 2

Factor $4$ out of $16 t^{2} - 60 t - 80$ .

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Step 2.1

Factor $4$ out of $16 t^{2}$ .

$4 (4 t^{2}) - 60 t - 80 = 0$

Step 2.2

Factor $4$ out of $- 60 t$ .

$4 (4 t^{2}) + 4 (- 15 t) - 80 = 0$

Step 2.3

Factor $4$ out of $- 80$ .

$4 (4 t^{2}) + 4 (- 15 t) + 4 (- 20) = 0$

Step 2.4

Factor $4$ out of $4 (4 t^{2}) + 4 (- 15 t)$ .

$4 (4 t^{2} - 15 t) + 4 (- 20) = 0$

Step 2.5

Factor $4$ out of $4 (4 t^{2} - 15 t) + 4 (- 20)$ .

$4 (4 t^{2} - 15 t - 20) = 0$

Step 3

Divide each term in $4 (4 t^{2} - 15 t - 20) = 0$ by $4$ and simplify.

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Step 3.1

Divide each term in $4 (4 t^{2} - 15 t - 20) = 0$ by $4$ .

$\frac{4 (4 t^{2} - 15 t - 20)}{4} = \frac{0}{4}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $4$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{4 (4 t^{2} - 15 t - 20)}{4} = \frac{0}{4}$

Step 3.2.1.2

Divide $4 t^{2} - 15 t - 20$ by $1$ .

$4 t^{2} - 15 t - 20 = \frac{0}{4}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $0$ by $4$ .

$4 t^{2} - 15 t - 20 = 0$

Step 4

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 5

Substitute the values $a = 4$ , $b = - 15$ , and $c = - 20$ into the quadratic formula and solve for $t$ .

$\frac{15 \pm \sqrt{{(- 15)}^{2} - 4 \cdot (4 \cdot - 20)}}{2 \cdot 4}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $- 15$ to the power of $2$ .

$t = \frac{15 \pm \sqrt{225 - 4 \cdot 4 \cdot - 20}}{2 \cdot 4}$

Step 6.1.2

Multiply $- 4 \cdot 4 \cdot - 20$ .

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Step 6.1.2.1

Multiply $- 4$ by $4$ .

$t = \frac{15 \pm \sqrt{225 - 16 \cdot - 20}}{2 \cdot 4}$

Step 6.1.2.2

Multiply $- 16$ by $- 20$ .

$t = \frac{15 \pm \sqrt{225 + 320}}{2 \cdot 4}$

Step 6.1.3

Add $225$ and $320$ .

$t = \frac{15 \pm \sqrt{545}}{2 \cdot 4}$

Step 6.2

Multiply $2$ by $4$ .

$t = \frac{15 \pm \sqrt{545}}{8}$

Step 7

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $- 15$ to the power of $2$ .

$t = \frac{15 \pm \sqrt{225 - 4 \cdot 4 \cdot - 20}}{2 \cdot 4}$

Step 7.1.2

Multiply $- 4 \cdot 4 \cdot - 20$ .

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Step 7.1.2.1

Multiply $- 4$ by $4$ .

$t = \frac{15 \pm \sqrt{225 - 16 \cdot - 20}}{2 \cdot 4}$

Step 7.1.2.2

Multiply $- 16$ by $- 20$ .

$t = \frac{15 \pm \sqrt{225 + 320}}{2 \cdot 4}$

Step 7.1.3

Add $225$ and $320$ .

$t = \frac{15 \pm \sqrt{545}}{2 \cdot 4}$

Step 7.2

Multiply $2$ by $4$ .

$t = \frac{15 \pm \sqrt{545}}{8}$

Step 7.3

Change the $\pm$ to $+$ .

$t = \frac{15 + \sqrt{545}}{8}$

Step 8

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Raise $- 15$ to the power of $2$ .

$t = \frac{15 \pm \sqrt{225 - 4 \cdot 4 \cdot - 20}}{2 \cdot 4}$

Step 8.1.2

Multiply $- 4 \cdot 4 \cdot - 20$ .

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Step 8.1.2.1

Multiply $- 4$ by $4$ .

$t = \frac{15 \pm \sqrt{225 - 16 \cdot - 20}}{2 \cdot 4}$

Step 8.1.2.2

Multiply $- 16$ by $- 20$ .

$t = \frac{15 \pm \sqrt{225 + 320}}{2 \cdot 4}$

Step 8.1.3

Add $225$ and $320$ .

$t = \frac{15 \pm \sqrt{545}}{2 \cdot 4}$

Step 8.2

Multiply $2$ by $4$ .

$t = \frac{15 \pm \sqrt{545}}{8}$

Step 8.3

Change the $\pm$ to $-$ .

$t = \frac{15 - \sqrt{545}}{8}$

Step 9

The final answer is the combination of both solutions.

$t = \frac{15 + \sqrt{545}}{8}, \frac{15 - \sqrt{545}}{8}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$t = \frac{15 + \sqrt{545}}{8}, \frac{15 - \sqrt{545}}{8}$

Decimal Form:

$t = 4.79315438 \dots, - 1.04315438 \dots$