Calculus Examples

$150 (8) = - 16 t^{2} + 150 t$

Step 1

Rewrite the equation as $- 16 t^{2} + 150 t = 150 (8)$ .

$- 16 t^{2} + 150 t = 150 (8)$

Step 2

Multiply $150$ by $8$ .

$- 16 t^{2} + 150 t = 1200$

Step 3

Subtract $1200$ from both sides of the equation.

$- 16 t^{2} + 150 t - 1200 = 0$

Step 4

Factor $- 2$ out of $- 16 t^{2} + 150 t - 1200$ .

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Step 4.1

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

$- 2 (8 t^{2}) + 150 t - 1200 = 0$

Step 4.2

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

$- 2 (8 t^{2}) - 2 (- 75 t) - 1200 = 0$

Step 4.3

Factor $- 2$ out of $- 1200$ .

$- 2 (8 t^{2}) - 2 (- 75 t) - 2 \cdot 600 = 0$

Step 4.4

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

$- 2 (8 t^{2} - 75 t) - 2 \cdot 600 = 0$

Step 4.5

Factor $- 2$ out of $- 2 (8 t^{2} - 75 t) - 2 (600)$ .

$- 2 (8 t^{2} - 75 t + 600) = 0$

Step 5

Divide each term in $- 2 (8 t^{2} - 75 t + 600) = 0$ by $- 2$ and simplify.

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Step 5.1

Divide each term in $- 2 (8 t^{2} - 75 t + 600) = 0$ by $- 2$ .

$\frac{- 2 (8 t^{2} - 75 t + 600)}{- 2} = \frac{0}{- 2}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $- 2$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{- 2 (8 t^{2} - 75 t + 600)}{- 2} = \frac{0}{- 2}$

Step 5.2.1.2

Divide $8 t^{2} - 75 t + 600$ by $1$ .

$8 t^{2} - 75 t + 600 = \frac{0}{- 2}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Divide $0$ by $- 2$ .

$8 t^{2} - 75 t + 600 = 0$

Step 6

Use the quadratic formula to find the solutions.

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

Step 7

Substitute the values $a = 8$ , $b = - 75$ , and $c = 600$ into the quadratic formula and solve for $t$ .

$\frac{75 \pm \sqrt{{(- 75)}^{2} - 4 \cdot (8 \cdot 600)}}{2 \cdot 8}$

Step 8

Simplify.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

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

$t = \frac{75 \pm \sqrt{5625 - 4 \cdot 8 \cdot 600}}{2 \cdot 8}$

Step 8.1.2

Multiply $- 4 \cdot 8 \cdot 600$ .

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Step 8.1.2.1

Multiply $- 4$ by $8$ .

$t = \frac{75 \pm \sqrt{5625 - 32 \cdot 600}}{2 \cdot 8}$

Step 8.1.2.2

Multiply $- 32$ by $600$ .

$t = \frac{75 \pm \sqrt{5625 - 19200}}{2 \cdot 8}$

Step 8.1.3

Subtract $19200$ from $5625$ .

$t = \frac{75 \pm \sqrt{- 13575}}{2 \cdot 8}$

Step 8.1.4

Rewrite $- 13575$ as $- 1 (13575)$ .

$t = \frac{75 \pm \sqrt{- 1 \cdot 13575}}{2 \cdot 8}$

Step 8.1.5

Rewrite $\sqrt{- 1 (13575)}$ as $\sqrt{- 1} \cdot \sqrt{13575}$ .

$t = \frac{75 \pm \sqrt{- 1} \cdot \sqrt{13575}}{2 \cdot 8}$

Step 8.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$t = \frac{75 \pm i \cdot \sqrt{13575}}{2 \cdot 8}$

Step 8.1.7

Rewrite $13575$ as $5^{2} \cdot 543$ .

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Step 8.1.7.1

Factor $25$ out of $13575$ .

$t = \frac{75 \pm i \cdot \sqrt{25 (543)}}{2 \cdot 8}$

Step 8.1.7.2

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

$t = \frac{75 \pm i \cdot \sqrt{5^{2} \cdot 543}}{2 \cdot 8}$

Step 8.1.8

Pull terms out from under the radical.

$t = \frac{75 \pm i \cdot (5 \sqrt{543})}{2 \cdot 8}$

Step 8.1.9

Move $5$ to the left of $i$ .

$t = \frac{75 \pm 5 i \sqrt{543}}{2 \cdot 8}$

Step 8.2

Multiply $2$ by $8$ .

$t = \frac{75 \pm 5 i \sqrt{543}}{16}$

Step 9

The final answer is the combination of both solutions.

$t = \frac{75 + 5 i \sqrt{543}}{16}, \frac{75 - 5 i \sqrt{543}}{16}$