Algebra Examples

$s = - 16 t^{2} + 96 t$

Step 1

Rewrite the equation as $- 16 t^{2} + 96 t = s$ .

$- 16 t^{2} + 96 t = s$

Step 2

Subtract $s$ from both sides of the equation.

$- 16 t^{2} + 96 t - s = 0$

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 $a = - 16$ , $b = 96$ , and $c = - s$ into the quadratic formula and solve for $t$ .

$\frac{- 96 \pm \sqrt{96^{2} - 4 \cdot (- 16 \cdot (- s))}}{2 \cdot - 16}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $96$ to the power of $2$ .

$t = \frac{- 96 \pm \sqrt{9216 - 4 \cdot - 16 \cdot (- 1 s)}}{2 \cdot - 16}$

Step 5.1.2

Multiply $- 4 \cdot - 16 \cdot - 1$ .

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Step 5.1.2.1

Multiply $- 4$ by $- 16$ .

$t = \frac{- 96 \pm \sqrt{9216 + 64 \cdot (- 1 s)}}{2 \cdot - 16}$

Step 5.1.2.2

Multiply $64$ by $- 1$ .

$t = \frac{- 96 \pm \sqrt{9216 - 64 s}}{2 \cdot - 16}$

Step 5.1.3

Factor $64$ out of $9216 - 64 s$ .

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Step 5.1.3.1

Factor $64$ out of $9216$ .

$t = \frac{- 96 \pm \sqrt{64 (144) - 64 s}}{2 \cdot - 16}$

Step 5.1.3.2

Factor $64$ out of $- 64 s$ .

$t = \frac{- 96 \pm \sqrt{64 (144) + 64 (- s)}}{2 \cdot - 16}$

Step 5.1.3.3

Factor $64$ out of $64 (144) + 64 (- s)$ .

$t = \frac{- 96 \pm \sqrt{64 (144 - s)}}{2 \cdot - 16}$

Step 5.1.4

Rewrite $64 (144 - s)$ as $8^{2} (12^{2} - s)$ .

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Step 5.1.4.1

Rewrite $64$ as $8^{2}$ .

$t = \frac{- 96 \pm \sqrt{8^{2} (144 - s)}}{2 \cdot - 16}$

Step 5.1.4.2

Rewrite $144$ as $12^{2}$ .

$t = \frac{- 96 \pm \sqrt{8^{2} (12^{2} - s)}}{2 \cdot - 16}$

Step 5.1.5

Pull terms out from under the radical.

$t = \frac{- 96 \pm 8 \sqrt{12^{2} - s}}{2 \cdot - 16}$

Step 5.1.6

Raise $12$ to the power of $2$ .

$t = \frac{- 96 \pm 8 \sqrt{144 - s}}{2 \cdot - 16}$

Step 5.2

Multiply $2$ by $- 16$ .

$t = \frac{- 96 \pm 8 \sqrt{144 - s}}{- 32}$

Step 5.3

Simplify $\frac{- 96 \pm 8 \sqrt{144 - s}}{- 32}$ .

$t = \frac{12 \pm \sqrt{144 - s}}{4}$

Step 6

The final answer is the combination of both solutions.

$t = \frac{12 + \sqrt{144 - s}}{4}$

$t = \frac{12 - \sqrt{144 - s}}{4}$