Pre-Algebra Examples

$60 = 70 t - 16 t^{2}$

Step 1

Rewrite the equation as $70 t - 16 t^{2} = 60$ .

$70 t - 16 t^{2} = 60$

Step 2

Subtract $60$ from both sides of the equation.

$70 t - 16 t^{2} - 60 = 0$

Step 3

Factor $- 2$ out of $70 t - 16 t^{2} - 60$ .

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Step 3.1

Reorder $70 t$ and $- 16 t^{2}$ .

$- 16 t^{2} + 70 t - 60 = 0$

Step 3.2

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

$- 2 (8 t^{2}) + 70 t - 60 = 0$

Step 3.3

Factor $- 2$ out of $70 t$ .

$- 2 (8 t^{2}) - 2 (- 35 t) - 60 = 0$

Step 3.4

Factor $- 2$ out of $- 60$ .

$- 2 (8 t^{2}) - 2 (- 35 t) - 2 \cdot 30 = 0$

Step 3.5

Factor $- 2$ out of $- 2 (8 t^{2}) - 2 (- 35 t)$ .

$- 2 (8 t^{2} - 35 t) - 2 \cdot 30 = 0$

Step 3.6

Factor $- 2$ out of $- 2 (8 t^{2} - 35 t) - 2 (30)$ .

$- 2 (8 t^{2} - 35 t + 30) = 0$

Step 4

Divide each term in $- 2 (8 t^{2} - 35 t + 30) = 0$ by $- 2$ and simplify.

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Step 4.1

Divide each term in $- 2 (8 t^{2} - 35 t + 30) = 0$ by $- 2$ .

$\frac{- 2 (8 t^{2} - 35 t + 30)}{- 2} = \frac{0}{- 2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $- 2$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{- 2 (8 t^{2} - 35 t + 30)}{- 2} = \frac{0}{- 2}$

Step 4.2.1.2

Divide $8 t^{2} - 35 t + 30$ by $1$ .

$8 t^{2} - 35 t + 30 = \frac{0}{- 2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $0$ by $- 2$ .

$8 t^{2} - 35 t + 30 = 0$

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 $a = 8$ , $b = - 35$ , and $c = 30$ into the quadratic formula and solve for $t$ .

$\frac{35 \pm \sqrt{{(- 35)}^{2} - 4 \cdot (8 \cdot 30)}}{2 \cdot 8}$

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 $2$ .

$t = \frac{35 \pm \sqrt{1225 - 4 \cdot 8 \cdot 30}}{2 \cdot 8}$

Step 7.1.2

Multiply $- 4 \cdot 8 \cdot 30$ .

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Step 7.1.2.1

Multiply $- 4$ by $8$ .

$t = \frac{35 \pm \sqrt{1225 - 32 \cdot 30}}{2 \cdot 8}$

Step 7.1.2.2

Multiply $- 32$ by $30$ .

$t = \frac{35 \pm \sqrt{1225 - 960}}{2 \cdot 8}$

Step 7.1.3

Subtract $960$ from $1225$ .

$t = \frac{35 \pm \sqrt{265}}{2 \cdot 8}$

Step 7.2

Multiply $2$ by $8$ .

$t = \frac{35 \pm \sqrt{265}}{16}$

Step 8

The final answer is the combination of both solutions.

$t = \frac{35 + \sqrt{265}}{16}, \frac{35 - \sqrt{265}}{16}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$t = \frac{35 + \sqrt{265}}{16}, \frac{35 - \sqrt{265}}{16}$

Decimal Form:

$t = 3.20492628 \dots, 1.17007371 \dots$