Algebra Examples

$- \frac{1}{2} x {(x^{2} - 9)}^{2}$

Step 1

Set $- \frac{1}{2} \cdot (x {(x^{2} - 9)}^{2})$ equal to $0$ .

$- \frac{1}{2} \cdot (x {(x^{2} - 9)}^{2}) = 0$

Step 2

Solve for $x$ .

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Step 2.1

Multiply both sides of the equation by $- 2$ .

$- 2 (- \frac{1}{2} \cdot (x {(x^{2} - 9)}^{2})) = - 2 \cdot 0$

Step 2.2

Simplify both sides of the equation.

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Step 2.2.1

Simplify the left side.

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Step 2.2.1.1

Simplify $- 2 (- \frac{1}{2} \cdot (x {(x^{2} - 9)}^{2}))$ .

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Step 2.2.1.1.1

Multiply $- \frac{1}{2} (x {(x^{2} - 9)}^{2})$ .

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Step 2.2.1.1.1.1

Combine $x$ and $\frac{1}{2}$ .

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

Step 2.2.1.1.1.2

Combine ${(x^{2} - 9)}^{2}$ and $\frac{x}{2}$ .

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

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Move the leading negative in $- \frac{{(x^{2} - 9)}^{2} x}{2}$ into the numerator.

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

Step 2.2.1.1.2.2

Factor $2$ out of $- 2$ .

$2 (- 1) \frac{- {(x^{2} - 9)}^{2} x}{2} = - 2 \cdot 0$

Step 2.2.1.1.2.3

Cancel the common factor.

$2 \cdot - 1 \frac{- {(x^{2} - 9)}^{2} x}{2} = - 2 \cdot 0$

Step 2.2.1.1.2.4

Rewrite the expression.

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

Step 2.2.1.1.3

Simplify the expression.

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Step 2.2.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 ({(x^{2} - 9)}^{2} x) = - 2 \cdot 0$

Step 2.2.1.1.3.2

Multiply ${(x^{2} - 9)}^{2}$ by $1$ .

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

Step 2.2.1.1.3.3

Reorder factors in ${(x^{2} - 9)}^{2} x$ .

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

Step 2.2.2

Simplify the right side.

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Step 2.2.2.1

Multiply $- 2$ by $0$ .

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

Step 2.3

Factor.

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Step 2.3.1

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

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

Step 2.3.2

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 $b = 3$ .

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

Step 2.3.3

Apply the product rule to $(x + 3) (x - 3)$ .

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

Step 2.4

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

$x = 0$

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

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

Step 2.5

Set $x$ equal to $0$ .

$x = 0$

Step 2.6

Set ${(x + 3)}^{2}$ equal to $0$ and solve for $x$ .

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Step 2.6.1

Set ${(x + 3)}^{2}$ equal to $0$ .

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

Step 2.6.2

Solve ${(x + 3)}^{2} = 0$ for $x$ .

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Step 2.6.2.1

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

$x + 3 = 0$

Step 2.6.2.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 2.7

Set ${(x - 3)}^{2}$ equal to $0$ and solve for $x$ .

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Step 2.7.1

Set ${(x - 3)}^{2}$ equal to $0$ .

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

Step 2.7.2

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

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Step 2.7.2.1

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

$x - 3 = 0$

Step 2.7.2.2

Add $3$ to both sides of the equation.

$x = 3$

Step 2.8

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

$x = 0$ (Multiplicity of $1$ )

$x = - 3$ (Multiplicity of $2$ )

$x = 3$ (Multiplicity of $2$ )

$x = 0$ (Multiplicity of $1$ )

$x = - 3$ (Multiplicity of $2$ )

$x = 3$ (Multiplicity of $2$ )

Step 3