Algebra Examples

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

Step 1

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

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

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

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

Step 2.1.2

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 $b = 2$ .

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

Step 2.1.3

Apply the product rule to $(x + 2) (x - 2)$ .

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

Step 2.2

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

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

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

Step 2.3

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

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Step 2.3.1

Set ${(x + 2)}^{2}$ equal to $0$ .

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

Step 2.3.2

Solve ${(x + 2)}^{2} = 0$ for $x$ .

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Step 2.3.2.1

Set the $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 2.3.2.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 2.4

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

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Step 2.4.1

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

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

Step 2.4.2

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

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Step 2.4.2.1

Set the $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 2.4.2.2

Add $2$ to both sides of the equation.

$x = 2$

Step 2.5

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

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

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

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

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

Step 3