Precalculus Examples

$x^{6} - 16 x^{4} = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Factor $x^{4}$ out of $x^{6} - 16 x^{4}$ .

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Step 1.1.1

Factor $x^{4}$ out of $x^{6}$ .

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

Step 1.1.2

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

$x^{4} x^{2} + x^{4} \cdot - 16 = 0$

Step 1.1.3

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

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

Step 1.2

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

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

Step 1.3

Factor.

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Step 1.3.1

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 $b = 4$ .

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

Step 1.3.2

Remove unnecessary parentheses.

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

Step 2

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

$x^{4} = 0$

$x + 4 = 0$

$x - 4 = 0$

Step 3

Set $x^{4}$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $x^{4}$ equal to $0$ .

$x^{4} = 0$

Step 3.2

Solve $x^{4} = 0$ for $x$ .

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Step 3.2.1

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

$x = \pm \sqrt[4]{0}$

Step 3.2.2

Simplify $\pm \sqrt[4]{0}$ .

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Step 3.2.2.1

Rewrite $0$ as $0^{4}$ .

$x = \pm \sqrt[4]{0^{4}}$

Step 3.2.2.2

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

$x = \pm 0$

Step 3.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 4

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

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Step 4.1

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

$x + 4 = 0$

Step 4.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 5

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

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Step 5.1

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

$x - 4 = 0$

Step 5.2

Add $4$ to both sides of the equation.

$x = 4$

Step 6

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

$x = 0, - 4, 4$