Finite Math Examples

$0 = - 16 t^{2} + 150 t + 800$

Step 1

Rewrite the equation as $- 16 t^{2} + 150 t + 800 = 0$ .

$- 16 t^{2} + 150 t + 800 = 0$

Step 2

Factor $- 2$ out of $- 16 t^{2} + 150 t + 800$ .

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Step 2.1

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

$- 2 (8 t^{2}) + 150 t + 800 = 0$

Step 2.2

Factor $- 2$ out of $150 t$ .

$- 2 (8 t^{2}) - 2 (- 75 t) + 800 = 0$

Step 2.3

Factor $- 2$ out of $800$ .

$- 2 (8 t^{2}) - 2 (- 75 t) - 2 \cdot - 400 = 0$

Step 2.4

Factor $- 2$ out of $- 2 (8 t^{2}) - 2 (- 75 t)$ .

$- 2 (8 t^{2} - 75 t) - 2 \cdot - 400 = 0$

Step 2.5

Factor $- 2$ out of $- 2 (8 t^{2} - 75 t) - 2 (- 400)$ .

$- 2 (8 t^{2} - 75 t - 400) = 0$

Step 3

Divide each term in $- 2 (8 t^{2} - 75 t - 400) = 0$ by $- 2$ and simplify.

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Step 3.1

Divide each term in $- 2 (8 t^{2} - 75 t - 400) = 0$ by $- 2$ .

$\frac{- 2 (8 t^{2} - 75 t - 400)}{- 2} = \frac{0}{- 2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $- 2$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{- 2 (8 t^{2} - 75 t - 400)}{- 2} = \frac{0}{- 2}$

Step 3.2.1.2

Divide $8 t^{2} - 75 t - 400$ by $1$ .

$8 t^{2} - 75 t - 400 = \frac{0}{- 2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $0$ by $- 2$ .

$8 t^{2} - 75 t - 400 = 0$

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 $a = 8$ , $b = - 75$ , and $c = - 400$ into the quadratic formula and solve for $t$ .

$\frac{75 \pm \sqrt{{(- 75)}^{2} - 4 \cdot (8 \cdot - 400)}}{2 \cdot 8}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $- 75$ to the power of $2$ .

$t = \frac{75 \pm \sqrt{5625 - 4 \cdot 8 \cdot - 400}}{2 \cdot 8}$

Step 6.1.2

Multiply $- 4 \cdot 8 \cdot - 400$ .

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Step 6.1.2.1

Multiply $- 4$ by $8$ .

$t = \frac{75 \pm \sqrt{5625 - 32 \cdot - 400}}{2 \cdot 8}$

Step 6.1.2.2

Multiply $- 32$ by $- 400$ .

$t = \frac{75 \pm \sqrt{5625 + 12800}}{2 \cdot 8}$

Step 6.1.3

Add $5625$ and $12800$ .

$t = \frac{75 \pm \sqrt{18425}}{2 \cdot 8}$

Step 6.1.4

Rewrite $18425$ as $5^{2} \cdot 737$ .

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Step 6.1.4.1

Factor $25$ out of $18425$ .

$t = \frac{75 \pm \sqrt{25 (737)}}{2 \cdot 8}$

Step 6.1.4.2

Rewrite $25$ as $5^{2}$ .

$t = \frac{75 \pm \sqrt{5^{2} \cdot 737}}{2 \cdot 8}$

Step 6.1.5

Pull terms out from under the radical.

$t = \frac{75 \pm 5 \sqrt{737}}{2 \cdot 8}$

Step 6.2

Multiply $2$ by $8$ .

$t = \frac{75 \pm 5 \sqrt{737}}{16}$

Step 7

The final answer is the combination of both solutions.

$t = \frac{75 + 5 \sqrt{737}}{16}, \frac{75 - 5 \sqrt{737}}{16}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$t = \frac{75 + 5 \sqrt{737}}{16}, \frac{75 - 5 \sqrt{737}}{16}$

Decimal Form:

$t = 13.17116997 \dots, - 3.79616997 \dots$