Precalculus Examples

$68 = 140 t - 16 t^{2}$

Step 1

Rewrite the equation as $140 t - 16 t^{2} = 68$ .

$140 t - 16 t^{2} = 68$

Step 2

Subtract $68$ from both sides of the equation.

$140 t - 16 t^{2} - 68 = 0$

Step 3

Factor $- 4$ out of $140 t - 16 t^{2} - 68$ .

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Step 3.1

Reorder $140 t$ and $- 16 t^{2}$ .

$- 16 t^{2} + 140 t - 68 = 0$

Step 3.2

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

$- 4 (4 t^{2}) + 140 t - 68 = 0$

Step 3.3

Factor $- 4$ out of $140 t$ .

$- 4 (4 t^{2}) - 4 (- 35 t) - 68 = 0$

Step 3.4

Factor $- 4$ out of $- 68$ .

$- 4 (4 t^{2}) - 4 (- 35 t) - 4 \cdot 17 = 0$

Step 3.5

Factor $- 4$ out of $- 4 (4 t^{2}) - 4 (- 35 t)$ .

$- 4 (4 t^{2} - 35 t) - 4 \cdot 17 = 0$

Step 3.6

Factor $- 4$ out of $- 4 (4 t^{2} - 35 t) - 4 (17)$ .

$- 4 (4 t^{2} - 35 t + 17) = 0$

Step 4

Divide each term in $- 4 (4 t^{2} - 35 t + 17) = 0$ by $- 4$ and simplify.

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Step 4.1

Divide each term in $- 4 (4 t^{2} - 35 t + 17) = 0$ by $- 4$ .

$\frac{- 4 (4 t^{2} - 35 t + 17)}{- 4} = \frac{0}{- 4}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $- 4$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{- 4 (4 t^{2} - 35 t + 17)}{- 4} = \frac{0}{- 4}$

Step 4.2.1.2

Divide $4 t^{2} - 35 t + 17$ by $1$ .

$4 t^{2} - 35 t + 17 = \frac{0}{- 4}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $0$ by $- 4$ .

$4 t^{2} - 35 t + 17 = 0$

Step 5

Use the quadratic formula to find the solutions.

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

Step 6

Substitute the values $a = 4$ , $b = - 35$ , and $c = 17$ into the quadratic formula and solve for $t$ .

$\frac{35 \pm \sqrt{{(- 35)}^{2} - 4 \cdot (4 \cdot 17)}}{2 \cdot 4}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $- 35$ to the power of $2$ .

$t = \frac{35 \pm \sqrt{1225 - 4 \cdot 4 \cdot 17}}{2 \cdot 4}$

Step 7.1.2

Multiply $- 4 \cdot 4 \cdot 17$ .

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Step 7.1.2.1

Multiply $- 4$ by $4$ .

$t = \frac{35 \pm \sqrt{1225 - 16 \cdot 17}}{2 \cdot 4}$

Step 7.1.2.2

Multiply $- 16$ by $17$ .

$t = \frac{35 \pm \sqrt{1225 - 272}}{2 \cdot 4}$

Step 7.1.3

Subtract $272$ from $1225$ .

$t = \frac{35 \pm \sqrt{953}}{2 \cdot 4}$

Step 7.2

Multiply $2$ by $4$ .

$t = \frac{35 \pm \sqrt{953}}{8}$

Step 8

The final answer is the combination of both solutions.

$t = \frac{35 + \sqrt{953}}{8}, \frac{35 - \sqrt{953}}{8}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$t = \frac{35 + \sqrt{953}}{8}, \frac{35 - \sqrt{953}}{8}$

Decimal Form:

$t = 8.23383726 \dots, 0.51616273 \dots$