Pre-Algebra Examples

$h = - 16 t^{2} + 48 t + 64$

Step 1

Rewrite the equation as $- 16 t^{2} + 48 t + 64 = h$ .

$- 16 t^{2} + 48 t + 64 = h$

Step 2

Subtract $h$ from both sides of the equation.

$- 16 t^{2} + 48 t + 64 - h = 0$

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 $a = - 16$ , $b = 48$ , and $c = 64 - h$ into the quadratic formula and solve for $t$ .

$\frac{- 48 \pm \sqrt{48^{2} - 4 \cdot (- 16 \cdot (64 - h))}}{2 \cdot - 16}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $48$ to the power of $2$ .

$t = \frac{- 48 \pm \sqrt{2304 - 4 \cdot - 16 \cdot (64 - h)}}{2 \cdot - 16}$

Step 5.1.2

Multiply $- 4$ by $- 16$ .

$t = \frac{- 48 \pm \sqrt{2304 + 64 \cdot (64 - h)}}{2 \cdot - 16}$

Step 5.1.3

Apply the distributive property.

$t = \frac{- 48 \pm \sqrt{2304 + 64 \cdot 64 + 64 (- h)}}{2 \cdot - 16}$

Step 5.1.4

Multiply $64$ by $64$ .

$t = \frac{- 48 \pm \sqrt{2304 + 4096 + 64 (- h)}}{2 \cdot - 16}$

Step 5.1.5

Multiply $- 1$ by $64$ .

$t = \frac{- 48 \pm \sqrt{2304 + 4096 - 64 h}}{2 \cdot - 16}$

Step 5.1.6

Add $2304$ and $4096$ .

$t = \frac{- 48 \pm \sqrt{6400 - 64 h}}{2 \cdot - 16}$

Step 5.1.7

Factor $64$ out of $6400 - 64 h$ .

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Step 5.1.7.1

Factor $64$ out of $6400$ .

$t = \frac{- 48 \pm \sqrt{64 (100) - 64 h}}{2 \cdot - 16}$

Step 5.1.7.2

Factor $64$ out of $- 64 h$ .

$t = \frac{- 48 \pm \sqrt{64 (100) + 64 (- h)}}{2 \cdot - 16}$

Step 5.1.7.3

Factor $64$ out of $64 (100) + 64 (- h)$ .

$t = \frac{- 48 \pm \sqrt{64 (100 - h)}}{2 \cdot - 16}$

Step 5.1.8

Rewrite $64 (100 - h)$ as $8^{2} (10^{2} - h)$ .

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Step 5.1.8.1

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

$t = \frac{- 48 \pm \sqrt{8^{2} (100 - h)}}{2 \cdot - 16}$

Step 5.1.8.2

Rewrite $100$ as $10^{2}$ .

$t = \frac{- 48 \pm \sqrt{8^{2} (10^{2} - h)}}{2 \cdot - 16}$

Step 5.1.9

Pull terms out from under the radical.

$t = \frac{- 48 \pm 8 \sqrt{10^{2} - h}}{2 \cdot - 16}$

Step 5.1.10

Raise $10$ to the power of $2$ .

$t = \frac{- 48 \pm 8 \sqrt{100 - h}}{2 \cdot - 16}$

Step 5.2

Multiply $2$ by $- 16$ .

$t = \frac{- 48 \pm 8 \sqrt{100 - h}}{- 32}$

Step 5.3

Simplify $\frac{- 48 \pm 8 \sqrt{100 - h}}{- 32}$ .

$t = \frac{6 \pm \sqrt{100 - h}}{4}$

Step 6

The final answer is the combination of both solutions.

$t = \frac{6 + \sqrt{100 - h}}{4}$

$t = \frac{6 - \sqrt{100 - h}}{4}$