Algebra Examples

$f (x) = x^{4} - 9 x^{2}$

Step 1

Set $x^{4} - 9 x^{2}$ equal to $0$ .

$x^{4} - 9 x^{2} = 0$

Step 2

Solve for $x$ .

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Step 2.1

Factor the left side of the equation.

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Step 2.1.1

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

${(x^{2})}^{2} - 9 x^{2} = 0$

Step 2.1.2

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$u^{2} - 9 u = 0$

Step 2.1.3

Factor $u$ out of $u^{2} - 9 u$ .

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Step 2.1.3.1

Factor $u$ out of $u^{2}$ .

$u \cdot u - 9 u = 0$

Step 2.1.3.2

Factor $u$ out of $- 9 u$ .

$u \cdot u + u \cdot - 9 = 0$

Step 2.1.3.3

Factor $u$ out of $u \cdot u + u \cdot - 9$ .

$u (u - 9) = 0$

Step 2.1.4

Replace all occurrences of $u$ with $x^{2}$ .

$x^{2} (x^{2} - 9) = 0$

Step 2.2

Factor.

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Step 2.2.1

Rewrite $9$ as $3^{2}$ .

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

Step 2.2.2

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 $b = 3$ .

$x^{2} ((x + 3) (x - 3)) = 0$

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 $0$ .

$x^{2} = 0$

$x + 3 = 0$

$x - 3 = 0$

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 $x + 3$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Step 2.5.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 2.6

Set $x - 3$ equal to $0$ and solve for $x$ .

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Step 2.6.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 2.6.2

Add $3$ to both sides of the equation.

$x = 3$

Step 2.7

The final solution is all the values that make $x^{2} (x^{2} - 9) = 0$ true. The multiplicity of a root is the number of times the root appears.

$x = 0$ (Multiplicity of $2$ )

$x = - 3$ (Multiplicity of $1$ )

$x = 3$ (Multiplicity of $1$ )

$x = 0$ (Multiplicity of $2$ )

$x = - 3$ (Multiplicity of $1$ )

$x = 3$ (Multiplicity of $1$ )

Step 3