Algebra Examples

$h (x) = \frac{1}{4} x^{3}$

Step 1

Combine $\frac{1}{4}$ and $x^{3}$ .

$h x = \frac{x^{3}}{4}$

Step 2

Subtract $\frac{x^{3}}{4}$ from both sides of the equation.

$h x - \frac{x^{3}}{4} = 0$

Step 3

Factor $x$ out of $h x - \frac{x^{3}}{4}$ .

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Step 3.1

Factor $x$ out of $h x$ .

$x h - \frac{x^{3}}{4} = 0$

Step 3.2

Factor $x$ out of $- \frac{x^{3}}{4}$ .

$x h + x (- \frac{x^{2}}{4}) = 0$

Step 3.3

Factor $x$ out of $x h + x (- \frac{x^{2}}{4})$ .

$x (h - \frac{x^{2}}{4}) = 0$

Step 4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$h - \frac{x^{2}}{4} = 0$

Step 5

Set $x$ equal to $0$ .

$x = 0$

Step 6

Set $h - \frac{x^{2}}{4}$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $h - \frac{x^{2}}{4}$ equal to $0$ .

$h - \frac{x^{2}}{4} = 0$

Step 6.2

Solve $h - \frac{x^{2}}{4} = 0$ for $x$ .

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Step 6.2.1

Subtract $h$ from both sides of the equation.

$- \frac{x^{2}}{4} = - h$

Step 6.2.2

Multiply both sides of the equation by $- 4$ .

$- 4 (- \frac{x^{2}}{4}) = - 4 (- h)$

Step 6.2.3

Simplify both sides of the equation.

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Step 6.2.3.1

Simplify the left side.

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Step 6.2.3.1.1

Simplify $- 4 (- \frac{x^{2}}{4})$ .

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Step 6.2.3.1.1.1

Cancel the common factor of $4$ .

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Step 6.2.3.1.1.1.1

Move the leading negative in $- \frac{x^{2}}{4}$ into the numerator.

$- 4 \frac{- x^{2}}{4} = - 4 (- h)$

Step 6.2.3.1.1.1.2

Factor $4$ out of $- 4$ .

$4 (- 1) \frac{- x^{2}}{4} = - 4 (- h)$

Step 6.2.3.1.1.1.3

Cancel the common factor.

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

Step 6.2.3.1.1.1.4

Rewrite the expression.

$- - x^{2} = - 4 (- h)$

Step 6.2.3.1.1.2

Multiply.

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Step 6.2.3.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 x^{2} = - 4 (- h)$

Step 6.2.3.1.1.2.2

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

$x^{2} = - 4 (- h)$

Step 6.2.3.2

Simplify the right side.

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Step 6.2.3.2.1

Multiply $- 1$ by $- 4$ .

$x^{2} = 4 h$

Step 6.2.4

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

$x = \pm \sqrt{4 h}$

Step 6.2.5

Simplify $\pm \sqrt{4 h}$ .

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Step 6.2.5.1

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

$x = \pm \sqrt{2^{2} h}$

Step 6.2.5.2

Pull terms out from under the radical.

$x = \pm 2 \sqrt{h}$

Step 6.2.6

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

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Step 6.2.6.1

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

$x = 2 \sqrt{h}$

Step 6.2.6.2

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

$x = - 2 \sqrt{h}$

Step 6.2.6.3

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

$x = 2 \sqrt{h}$

$x = - 2 \sqrt{h}$

$x = 2 \sqrt{h}$

$x = - 2 \sqrt{h}$

$x = 2 \sqrt{h}$

$x = - 2 \sqrt{h}$

$x = 2 \sqrt{h}$

$x = - 2 \sqrt{h}$

Step 7

The final solution is all the values that make $x (h - \frac{x^{2}}{4}) = 0$ true.

$x = 0$

$x = 2 \sqrt{h}$

$x = - 2 \sqrt{h}$

Step 8

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.

$f (h) = 0, 2 \sqrt{h}, - 2 \sqrt{h}$