Algebra Examples

$h = - 16 x^{2} + 24$

Step 1

Rewrite the equation as $- 16 x^{2} + 24 = h$ .

$- 16 x^{2} + 24 = h$

Step 2

Subtract $24$ from both sides of the equation.

$- 16 x^{2} = h - 24$

Step 3

Divide each term in $- 16 x^{2} = h - 24$ by $- 16$ and simplify.

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Step 3.1

Divide each term in $- 16 x^{2} = h - 24$ by $- 16$ .

$\frac{- 16 x^{2}}{- 16} = \frac{h}{- 16} + \frac{- 24}{- 16}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $- 16$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{- 16 x^{2}}{- 16} = \frac{h}{- 16} + \frac{- 24}{- 16}$

Step 3.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{h}{- 16} + \frac{- 24}{- 16}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Move the negative in front of the fraction.

$x^{2} = - \frac{h}{16} + \frac{- 24}{- 16}$

Step 3.3.1.2

Cancel the common factor of $- 24$ and $- 16$ .

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Step 3.3.1.2.1

Factor $- 8$ out of $- 24$ .

$x^{2} = - \frac{h}{16} + \frac{- 8 (3)}{- 16}$

Step 3.3.1.2.2

Cancel the common factors.

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Step 3.3.1.2.2.1

Factor $- 8$ out of $- 16$ .

$x^{2} = - \frac{h}{16} + \frac{- 8 \cdot 3}{- 8 \cdot 2}$

Step 3.3.1.2.2.2

Cancel the common factor.

$x^{2} = - \frac{h}{16} + \frac{- 8 \cdot 3}{- 8 \cdot 2}$

Step 3.3.1.2.2.3

Rewrite the expression.

$x^{2} = - \frac{h}{16} + \frac{3}{2}$

Step 4

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{- \frac{h}{16} + \frac{3}{2}}$

Step 5

Simplify $\pm \sqrt{- \frac{h}{16} + \frac{3}{2}}$ .

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Step 5.1

To write $\frac{3}{2}$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$x = \pm \sqrt{- \frac{h}{16} + \frac{3}{2} \cdot \frac{8}{8}}$

Step 5.2

Write each expression with a common denominator of $16$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.2.1

Multiply $\frac{3}{2}$ by $\frac{8}{8}$ .

$x = \pm \sqrt{- \frac{h}{16} + \frac{3 \cdot 8}{2 \cdot 8}}$

Step 5.2.2

Multiply $2$ by $8$ .

$x = \pm \sqrt{- \frac{h}{16} + \frac{3 \cdot 8}{16}}$

Step 5.3

Combine the numerators over the common denominator.

$x = \pm \sqrt{\frac{- h + 3 \cdot 8}{16}}$

Step 5.4

Multiply $3$ by $8$ .

$x = \pm \sqrt{\frac{- h + 24}{16}}$

Step 5.5

Rewrite $\frac{- h + 24}{16}$ as ${(\frac{1}{4})}^{2} (- h + 24)$ .

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Step 5.5.1

Factor the perfect power $1^{2}$ out of $- h + 24$ .

$x = \pm \sqrt{\frac{1^{2} (- h + 24)}{16}}$

Step 5.5.2

Factor the perfect power $4^{2}$ out of $16$ .

$x = \pm \sqrt{\frac{1^{2} (- h + 24)}{4^{2} \cdot 1}}$

Step 5.5.3

Rearrange the fraction $\frac{1^{2} (- h + 24)}{4^{2} \cdot 1}$ .

$x = \pm \sqrt{{(\frac{1}{4})}^{2} (- h + 24)}$

Step 5.6

Pull terms out from under the radical.

$x = \pm \frac{1}{4} \sqrt{- h + 24}$

Step 5.7

Combine $\frac{1}{4}$ and $\sqrt{- h + 24}$ .

$x = \pm \frac{\sqrt{- h + 24}}{4}$

Step 6

The complete solution is the result of both the positive and negative portions of the solution.

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Step 6.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \frac{\sqrt{- h + 24}}{4}$

Step 6.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \frac{\sqrt{- h + 24}}{4}$

Step 6.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{\sqrt{- h + 24}}{4}$

$x = - \frac{\sqrt{- h + 24}}{4}$

$x = \frac{\sqrt{- h + 24}}{4}$

$x = - \frac{\sqrt{- h + 24}}{4}$

Step 7

To rewrite as a function of $h$ , write the equation so that $x$ is by itself on one side of the equal sign and an expression involving only $h$ is on the other side.

$f (h) = \frac{\sqrt{- h + 24}}{4}, - \frac{\sqrt{- h + 24}}{4}$