Precalculus Examples

$\sqrt{x^{4} - 16 x^{2}}$

Step 1

Set the radicand in $\sqrt{x^{4} - 16 x^{2}}$ greater than or equal to $0$ to find where the expression is defined.

$x^{4} - 16 x^{2} \geq 0$

Step 2

Solve for $x$ .

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Step 2.1

Convert the inequality to an equation.

$x^{4} - 16 x^{2} = 0$

Step 2.2

Factor the left side of the equation.

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Step 2.2.1

Factor $x^{2}$ out of $x^{4} - 16 x^{2}$ .

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Step 2.2.1.1

Factor $x^{2}$ out of $x^{4}$ .

$x^{2} x^{2} - 16 x^{2} = 0$

Step 2.2.1.2

Factor $x^{2}$ out of $- 16 x^{2}$ .

$x^{2} x^{2} + x^{2} \cdot - 16 = 0$

Step 2.2.1.3

Factor $x^{2}$ out of $x^{2} x^{2} + x^{2} \cdot - 16$ .

$x^{2} (x^{2} - 16) = 0$

Step 2.2.2

Rewrite $16$ as $4^{2}$ .

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

Step 2.2.3

Factor.

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Step 2.2.3.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 4$ .

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

Step 2.2.3.2

Remove unnecessary parentheses.

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

Step 2.3

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

$x^{2} = 0$

$x + 4 = 0$

$x - 4 = 0$

Step 2.4

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

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Step 2.4.1

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

$x^{2} = 0$

Step 2.4.2

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

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Step 2.4.2.1

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

$x = \pm \sqrt{0}$

Step 2.4.2.2

Simplify $\pm \sqrt{0}$ .

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Step 2.4.2.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 2.4.2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Step 2.4.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 2.5

Set $x + 4$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Step 2.5.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 2.6

Set $x - 4$ equal to $0$ and solve for $x$ .

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Step 2.6.1

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 2.6.2

Add $4$ to both sides of the equation.

$x = 4$

Step 2.7

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

$x = 0, - 4, 4$

Step 2.8

Use each root to create test intervals.

$x < - 4$

$- 4 < x < 0$

$0 < x < 4$

$x > 4$

Step 2.9

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 2.9.1

Test a value on the interval $x < - 4$ to see if it makes the inequality true.

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Step 2.9.1.1

Choose a value on the interval $x < - 4$ and see if this value makes the original inequality true.

$x = - 6$

Step 2.9.1.2

Replace $x$ with $- 6$ in the original inequality.

${(- 6)}^{4} - 16 {(- 6)}^{2} \geq 0$

Step 2.9.1.3

The left side $720$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 2.9.2

Test a value on the interval $- 4 < x < 0$ to see if it makes the inequality true.

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Step 2.9.2.1

Choose a value on the interval $- 4 < x < 0$ and see if this value makes the original inequality true.

$x = - 2$

Step 2.9.2.2

Replace $x$ with $- 2$ in the original inequality.

${(- 2)}^{4} - 16 {(- 2)}^{2} \geq 0$

Step 2.9.2.3

The left side $- 48$ is less than the right side $0$ , which means that the given statement is false.

False

Step 2.9.3

Test a value on the interval $0 < x < 4$ to see if it makes the inequality true.

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Step 2.9.3.1

Choose a value on the interval $0 < x < 4$ and see if this value makes the original inequality true.

$x = 2$

Step 2.9.3.2

Replace $x$ with $2$ in the original inequality.

${(2)}^{4} - 16 {(2)}^{2} \geq 0$

Step 2.9.3.3

The left side $- 48$ is less than the right side $0$ , which means that the given statement is false.

False

Step 2.9.4

Test a value on the interval $x > 4$ to see if it makes the inequality true.

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Step 2.9.4.1

Choose a value on the interval $x > 4$ and see if this value makes the original inequality true.

$x = 6$

Step 2.9.4.2

Replace $x$ with $6$ in the original inequality.

${(6)}^{4} - 16 {(6)}^{2} \geq 0$

Step 2.9.4.3

The left side $720$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 2.9.5

Compare the intervals to determine which ones satisfy the original inequality.

$x < - 4$ True

$- 4 < x < 0$ False

$0 < x < 4$ False

$x > 4$ True

$x < - 4$ True

$- 4 < x < 0$ False

$0 < x < 4$ False

$x > 4$ True

Step 2.10

The solution consists of all of the true intervals.

$x \leq - 4$ or $x \geq 4$

Step 3

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(- \infty, - 4] \cup [4, \infty)$

Set-Builder Notation:

${x | x \leq - 4, x \geq 4}$

Step 4