Calculus Examples

$6 = 12 t - 0.83 t^{2}$

Step 1

Rewrite the equation as $12 t - 0.83 t^{2} = 6$ .

$12 t - 0.83 t^{2} = 6$

Step 2

Subtract $6$ from both sides of the equation.

$12 t - 0.83 t^{2} - 6 = 0$

Step 3

Factor $- 0.01$ out of $12 t - 0.83 t^{2} - 6$ .

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Step 3.1

Reorder $12 t$ and $- 0.83 t^{2}$ .

$- 0.83 t^{2} + 12 t - 6 = 0$

Step 3.2

Factor $- 0.01$ out of $- 0.83 t^{2}$ .

$- 0.01 (83 t^{2}) + 12 t - 6 = 0$

Step 3.3

Factor $- 0.01$ out of $12 t$ .

$- 0.01 (83 t^{2}) - 0.01 (- 1200 t) - 6 = 0$

Step 3.4

Factor $- 0.01$ out of $- 6$ .

$- 0.01 (83 t^{2}) - 0.01 (- 1200 t) - 0.01 \cdot 600 = 0$

Step 3.5

Factor $- 0.01$ out of $- 0.01 (83 t^{2}) - 0.01 (- 1200 t)$ .

$- 0.01 (83 t^{2} - 1200 t) - 0.01 \cdot 600 = 0$

Step 3.6

Factor $- 0.01$ out of $- 0.01 (83 t^{2} - 1200 t) - 0.01 (600)$ .

$- 0.01 (83 t^{2} - 1200 t + 600) = 0$

Step 4

Divide each term in $- 0.01 (83 t^{2} - 1200 t + 600) = 0$ by $- 0.01$ and simplify.

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Step 4.1

Divide each term in $- 0.01 (83 t^{2} - 1200 t + 600) = 0$ by $- 0.01$ .

$\frac{- 0.01 (83 t^{2} - 1200 t + 600)}{- 0.01} = \frac{0}{- 0.01}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $- 0.01$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{- 0.01 (83 t^{2} - 1200 t + 600)}{- 0.01} = \frac{0}{- 0.01}$

Step 4.2.1.2

Divide $83 t^{2} - 1200 t + 600$ by $1$ .

$83 t^{2} - 1200 t + 600 = \frac{0}{- 0.01}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $0$ by $- 0.01$ .

$83 t^{2} - 1200 t + 600 = 0$

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 $a = 83$ , $b = - 1200$ , and $c = 600$ into the quadratic formula and solve for $t$ .

$\frac{1200 \pm \sqrt{{(- 1200)}^{2} - 4 \cdot (83 \cdot 600)}}{2 \cdot 83}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $- 1200$ to the power of $2$ .

$t = \frac{1200 \pm \sqrt{1440000 - 4 \cdot 83 \cdot 600}}{2 \cdot 83}$

Step 7.1.2

Multiply $- 4 \cdot 83 \cdot 600$ .

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Step 7.1.2.1

Multiply $- 4$ by $83$ .

$t = \frac{1200 \pm \sqrt{1440000 - 332 \cdot 600}}{2 \cdot 83}$

Step 7.1.2.2

Multiply $- 332$ by $600$ .

$t = \frac{1200 \pm \sqrt{1440000 - 199200}}{2 \cdot 83}$

Step 7.1.3

Subtract $199200$ from $1440000$ .

$t = \frac{1200 \pm \sqrt{1240800}}{2 \cdot 83}$

Step 7.1.4

Rewrite $1240800$ as $20^{2} \cdot 3102$ .

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Step 7.1.4.1

Factor $400$ out of $1240800$ .

$t = \frac{1200 \pm \sqrt{400 (3102)}}{2 \cdot 83}$

Step 7.1.4.2

Rewrite $400$ as $20^{2}$ .

$t = \frac{1200 \pm \sqrt{20^{2} \cdot 3102}}{2 \cdot 83}$

Step 7.1.5

Pull terms out from under the radical.

$t = \frac{1200 \pm 20 \sqrt{3102}}{2 \cdot 83}$

Step 7.2

Multiply $2$ by $83$ .

$t = \frac{1200 \pm 20 \sqrt{3102}}{166}$

Step 7.3

Simplify $\frac{1200 \pm 20 \sqrt{3102}}{166}$ .

$t = \frac{600 \pm 10 \sqrt{3102}}{83}$

Step 8

The final answer is the combination of both solutions.

$t = \frac{600 + 10 \sqrt{3102}}{83}, \frac{600 - 10 \sqrt{3102}}{83}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$t = \frac{600 + 10 \sqrt{3102}}{83}, \frac{600 - 10 \sqrt{3102}}{83}$

Decimal Form:

$t = 13.93922906 \dots, 0.51860225 \dots$