Precalculus Examples

$f (x) = \frac{1}{2} x^{3} - x$

Step 1

Set $\frac{1}{2} x^{3} - x$ equal to $0$ .

$\frac{1}{2} x^{3} - x = 0$

Step 2

Solve for $x$ .

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Step 2.1

Combine $\frac{1}{2}$ and $x^{3}$ .

$\frac{x^{3}}{2} - x = 0$

Step 2.2

Multiply each term in $\frac{x^{3}}{2} - x = 0$ by $2$ to eliminate the fractions.

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Step 2.2.1

Multiply each term in $\frac{x^{3}}{2} - x = 0$ by $2$ .

$\frac{x^{3}}{2} \cdot 2 - x \cdot 2 = 0 \cdot 2$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Simplify each term.

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Step 2.2.2.1.1

Cancel the common factor of $2$ .

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Step 2.2.2.1.1.1

Cancel the common factor.

$\frac{x^{3}}{2} \cdot 2 - x \cdot 2 = 0 \cdot 2$

Step 2.2.2.1.1.2

Rewrite the expression.

$x^{3} - x \cdot 2 = 0 \cdot 2$

Step 2.2.2.1.2

Multiply $2$ by $- 1$ .

$x^{3} - 2 x = 0 \cdot 2$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Multiply $0$ by $2$ .

$x^{3} - 2 x = 0$

Step 2.3

Factor $x$ out of $x^{3} - 2 x$ .

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Step 2.3.1

Factor $x$ out of $x^{3}$ .

$x \cdot x^{2} - 2 x = 0$

Step 2.3.2

Factor $x$ out of $- 2 x$ .

$x \cdot x^{2} + x \cdot - 2 = 0$

Step 2.3.3

Factor $x$ out of $x \cdot x^{2} + x \cdot - 2$ .

$x (x^{2} - 2) = 0$

Step 2.4

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

$x = 0$

$x^{2} - 2 = 0$

Step 2.5

Set $x$ equal to $0$ .

$x = 0$

Step 2.6

Set $x^{2} - 2$ equal to $0$ and solve for $x$ .

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Step 2.6.1

Set $x^{2} - 2$ equal to $0$ .

$x^{2} - 2 = 0$

Step 2.6.2

Solve $x^{2} - 2 = 0$ for $x$ .

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Step 2.6.2.1

Add $2$ to both sides of the equation.

$x^{2} = 2$

Step 2.6.2.2

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

$x = \pm \sqrt{2}$

Step 2.6.2.3

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

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Step 2.6.2.3.1

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

$x = \sqrt{2}$

Step 2.6.2.3.2

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

$x = - \sqrt{2}$

Step 2.6.2.3.3

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

$x = \sqrt{2}, - \sqrt{2}$

Step 2.7

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

$x = 0, \sqrt{2}, - \sqrt{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = 0, \sqrt{2}, - \sqrt{2}$

Decimal Form:

$x = 0, 1.41421356 \dots, - 1.41421356 \dots$

Step 4