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