Algebra Examples

f (x) = \frac{x^{3} - 9 x}{3 x^{2} - 6 x - 9}

$f (x) = \frac{x^{3} - 9 x}{3 x^{2} - 6 x - 9}$

Step 1

Set

\frac{x^{3} - 9 x}{3 x^{2} - 6 x - 9}

$\frac{x^{3} - 9 x}{3 x^{2} - 6 x - 9}$ equal to

0

$0$ .

\frac{x^{3} - 9 x}{3 x^{2} - 6 x - 9} = 0

$\frac{x^{3} - 9 x}{3 x^{2} - 6 x - 9} = 0$

Step 2

Solve for

x

$x$ .

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Step 2.1

Set the numerator equal to zero.

x^{3} - 9 x = 0

$x^{3} - 9 x = 0$

Step 2.2

Solve the equation for

x

$x$ .

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Step 2.2.1

Factor the left side of the equation.

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Step 2.2.1.1

Factor

x

$x$ out of

x^{3} - 9 x

$x^{3} - 9 x$ .

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Step 2.2.1.1.1

Factor

x

$x$ out of

x^{3}

$x^{3}$ .

x \cdot x^{2} - 9 x = 0

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

Step 2.2.1.1.2

Factor

x

$x$ out of

- 9 x

$- 9 x$ .

x \cdot x^{2} + x \cdot - 9 = 0

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

Step 2.2.1.1.3

Factor

x

$x$ out of

x \cdot x^{2} + x \cdot - 9

$x \cdot x^{2} + x \cdot - 9$ .

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

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

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

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

Step 2.2.1.2

Rewrite

9

$9$ as

3^{2}

$3^{2}$ .

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

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

Step 2.2.1.3

Factor.

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Step 2.2.1.3.1

Since both terms are perfect squares, factor using the difference of squares formula,

a^{2} - b^{2} = (a + b) (a - b)

$a^{2} - b^{2} = (a + b) (a - b)$ where

a = x

$a = x$ and

b = 3

$b = 3$ .

x ((x + 3) (x - 3)) = 0

$x ((x + 3) (x - 3)) = 0$

Step 2.2.1.3.2

Remove unnecessary parentheses.

x (x + 3) (x - 3) = 0

$x (x + 3) (x - 3) = 0$

x (x + 3) (x - 3) = 0

$x (x + 3) (x - 3) = 0$

x (x + 3) (x - 3) = 0

$x (x + 3) (x - 3) = 0$

Step 2.2.2

If any individual factor on the left side of the equation is equal to

0

$0$ , the entire expression will be equal to

0

$0$ .

x = 0

$x = 0$

x + 3 = 0

$x + 3 = 0$

x - 3 = 0

$x - 3 = 0$

Step 2.2.3

Set

x

$x$ equal to

0

$0$ .

x = 0

$x = 0$

Step 2.2.4

Set

x + 3

$x + 3$ equal to

0

$0$ and solve for

x

$x$ .

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Step 2.2.4.1

Set

x + 3

$x + 3$ equal to

0

$0$ .

x + 3 = 0

$x + 3 = 0$

Step 2.2.4.2

Subtract

3

$3$ from both sides of the equation.

x = - 3

$x = - 3$

x = - 3

$x = - 3$

Step 2.2.5

Set

x - 3

$x - 3$ equal to

0

$0$ and solve for

x

$x$ .

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Step 2.2.5.1

Set

x - 3

$x - 3$ equal to

0

$0$ .

x - 3 = 0

$x - 3 = 0$

Step 2.2.5.2

Add

3

$3$ to both sides of the equation.

x = 3

$x = 3$

x = 3

$x = 3$

Step 2.2.6

The final solution is all the values that make

x (x + 3) (x - 3) = 0

$x (x + 3) (x - 3) = 0$ true.

x = 0, - 3, 3

$x = 0, - 3, 3$

x = 0, - 3, 3

$x = 0, - 3, 3$

Step 2.3

Exclude the solutions that do not make

\frac{x^{3} - 9 x}{3 x^{2} - 6 x - 9} = 0

$\frac{x^{3} - 9 x}{3 x^{2} - 6 x - 9} = 0$ true.

x = 0, - 3

$x = 0, - 3$

x = 0, - 3

$x = 0, - 3$

Step 3

[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$