Basic Math Examples

$- 16 t^{2} - 22 t + 100 = h$

Step 1

Subtract $h$ from both sides of the equation.

$- 16 t^{2} - 22 t + 100 - h = 0$

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

Substitute the values $a = - 16$ , $b = - 22$ , and $c = 100 - h$ into the quadratic formula and solve for $t$ .

$\frac{22 \pm \sqrt{{(- 22)}^{2} - 4 \cdot (- 16 \cdot (100 - h))}}{2 \cdot - 16}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $- 22$ to the power of $2$ .

$t = \frac{22 \pm \sqrt{484 - 4 \cdot - 16 \cdot (100 - h)}}{2 \cdot - 16}$

Step 4.1.2

Multiply $- 4$ by $- 16$ .

$t = \frac{22 \pm \sqrt{484 + 64 \cdot (100 - h)}}{2 \cdot - 16}$

Step 4.1.3

Apply the distributive property.

$t = \frac{22 \pm \sqrt{484 + 64 \cdot 100 + 64 (- h)}}{2 \cdot - 16}$

Step 4.1.4

Multiply $64$ by $100$ .

$t = \frac{22 \pm \sqrt{484 + 6400 + 64 (- h)}}{2 \cdot - 16}$

Step 4.1.5

Multiply $- 1$ by $64$ .

$t = \frac{22 \pm \sqrt{484 + 6400 - 64 h}}{2 \cdot - 16}$

Step 4.1.6

Add $484$ and $6400$ .

$t = \frac{22 \pm \sqrt{6884 - 64 h}}{2 \cdot - 16}$

Step 4.1.7

Factor $4$ out of $6884 - 64 h$ .

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Step 4.1.7.1

Factor $4$ out of $6884$ .

$t = \frac{22 \pm \sqrt{4 (1721) - 64 h}}{2 \cdot - 16}$

Step 4.1.7.2

Factor $4$ out of $- 64 h$ .

$t = \frac{22 \pm \sqrt{4 (1721) + 4 (- 16 h)}}{2 \cdot - 16}$

Step 4.1.7.3

Factor $4$ out of $4 (1721) + 4 (- 16 h)$ .

$t = \frac{22 \pm \sqrt{4 (1721 - 16 h)}}{2 \cdot - 16}$

Step 4.1.8

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

$t = \frac{22 \pm \sqrt{2^{2} (1721 - 16 h)}}{2 \cdot - 16}$

Step 4.1.9

Pull terms out from under the radical.

$t = \frac{22 \pm 2 \sqrt{1721 - 16 h}}{2 \cdot - 16}$

Step 4.2

Multiply $2$ by $- 16$ .

$t = \frac{22 \pm 2 \sqrt{1721 - 16 h}}{- 32}$

Step 4.3

Simplify $\frac{22 \pm 2 \sqrt{1721 - 16 h}}{- 32}$ .

$t = \frac{11 \pm \sqrt{1721 - 16 h}}{- 16}$

Step 4.4

Move the negative in front of the fraction.

$t = - \frac{11 \pm \sqrt{1721 - 16 h}}{16}$

Step 5

The final answer is the combination of both solutions.

$t = - \frac{11 + \sqrt{1721 - 16 h}}{16}$

$t = - \frac{11 - \sqrt{1721 - 16 h}}{16}$