Finite Math Examples

$s = 120 t - 3 t^{2}$

Step 1

Rewrite the equation as $120 t - 3 t^{2} = s$ .

$120 t - 3 t^{2} = s$

Step 2

Subtract $s$ from both sides of the equation.

$120 t - 3 t^{2} - s = 0$

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

Substitute the values $a = - 3$ , $b = 120$ , and $c = - s$ into the quadratic formula and solve for $t$ .

$\frac{- 120 \pm \sqrt{120^{2} - 4 \cdot (- 3 \cdot (- s))}}{2 \cdot - 3}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $120$ to the power of $2$ .

$t = \frac{- 120 \pm \sqrt{14400 - 4 \cdot - 3 \cdot (- 1 s)}}{2 \cdot - 3}$

Step 5.1.2

Multiply $- 4 \cdot - 3 \cdot - 1$ .

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Step 5.1.2.1

Multiply $- 4$ by $- 3$ .

$t = \frac{- 120 \pm \sqrt{14400 + 12 \cdot (- 1 s)}}{2 \cdot - 3}$

Step 5.1.2.2

Multiply $12$ by $- 1$ .

$t = \frac{- 120 \pm \sqrt{14400 - 12 s}}{2 \cdot - 3}$

Step 5.1.3

Factor $12$ out of $14400 - 12 s$ .

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Step 5.1.3.1

Factor $12$ out of $14400$ .

$t = \frac{- 120 \pm \sqrt{12 (1200) - 12 s}}{2 \cdot - 3}$

Step 5.1.3.2

Factor $12$ out of $- 12 s$ .

$t = \frac{- 120 \pm \sqrt{12 (1200) + 12 (- s)}}{2 \cdot - 3}$

Step 5.1.3.3

Factor $12$ out of $12 (1200) + 12 (- s)$ .

$t = \frac{- 120 \pm \sqrt{12 (1200 - s)}}{2 \cdot - 3}$

Step 5.1.4

Rewrite $12 (1200 - s)$ as $2^{2} \cdot (3 (1200 - s))$ .

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Step 5.1.4.1

Factor $4$ out of $12$ .

$t = \frac{- 120 \pm \sqrt{4 (3) (1200 - s)}}{2 \cdot - 3}$

Step 5.1.4.2

Rewrite $4$ as $2^{2}$ .

$t = \frac{- 120 \pm \sqrt{2^{2} \cdot (3 (1200 - s))}}{2 \cdot - 3}$

Step 5.1.4.3

Add parentheses.

$t = \frac{- 120 \pm \sqrt{2^{2} \cdot (3 (1200 - s))}}{2 \cdot - 3}$

Step 5.1.5

Pull terms out from under the radical.

$t = \frac{- 120 \pm 2 \sqrt{3 (1200 - s)}}{2 \cdot - 3}$

Step 5.2

Multiply $2$ by $- 3$ .

$t = \frac{- 120 \pm 2 \sqrt{3 (1200 - s)}}{- 6}$

Step 5.3

Simplify $\frac{- 120 \pm 2 \sqrt{3 (1200 - s)}}{- 6}$ .

$t = \frac{60 \pm \sqrt{3 (1200 - s)}}{3}$

Step 6

The final answer is the combination of both solutions.

$t = \frac{60 + \sqrt{3 (1200 - s)}}{3}$

$t = \frac{60 - \sqrt{3 (1200 - s)}}{3}$