Precalculus Examples

$\frac{\frac{1}{x^{2}} - \frac{1}{9}}{x - 3}$

Step 1

Multiply the numerator and denominator of the fraction by $9 x^{2}$ .

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Step 1.1

Multiply $\frac{\frac{1}{x^{2}} - \frac{1}{9}}{x - 3}$ by $\frac{9 x^{2}}{9 x^{2}}$ .

$\frac{9 x^{2}}{9 x^{2}} \cdot \frac{\frac{1}{x^{2}} - \frac{1}{9}}{x - 3}$

Step 1.2

Combine.

$\frac{9 x^{2} (\frac{1}{x^{2}} - \frac{1}{9})}{9 x^{2} (x - 3)}$

Step 2

Apply the distributive property.

$\frac{9 x^{2} \frac{1}{x^{2}} + 9 x^{2} (- \frac{1}{9})}{9 x^{2} x + 9 x^{2} \cdot - 3}$

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $x^{2}$ .

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Step 3.1.1

Factor $x^{2}$ out of $9 x^{2}$ .

$\frac{x^{2} \cdot 9 \frac{1}{x^{2}} + 9 x^{2} (- \frac{1}{9})}{9 x^{2} x + 9 x^{2} \cdot - 3}$

Step 3.1.2

Cancel the common factor.

$\frac{x^{2} \cdot 9 \frac{1}{x^{2}} + 9 x^{2} (- \frac{1}{9})}{9 x^{2} x + 9 x^{2} \cdot - 3}$

Step 3.1.3

Rewrite the expression.

$\frac{9 + 9 x^{2} (- \frac{1}{9})}{9 x^{2} x + 9 x^{2} \cdot - 3}$

Step 3.2

Cancel the common factor of $9$ .

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Step 3.2.1

Move the leading negative in $- \frac{1}{9}$ into the numerator.

$\frac{9 + 9 x^{2} \frac{- 1}{9}}{9 x^{2} x + 9 x^{2} \cdot - 3}$

Step 3.2.2

Factor $9$ out of $9 x^{2}$ .

$\frac{9 + 9 (x^{2}) \frac{- 1}{9}}{9 x^{2} x + 9 x^{2} \cdot - 3}$

Step 3.2.3

Cancel the common factor.

$\frac{9 + 9 x^{2} \frac{- 1}{9}}{9 x^{2} x + 9 x^{2} \cdot - 3}$

Step 3.2.4

Rewrite the expression.

$\frac{9 + x^{2} \cdot - 1}{9 x^{2} x + 9 x^{2} \cdot - 3}$

Step 4

Simplify the numerator.

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Step 4.1

Move $- 1$ to the left of $x^{2}$ .

$\frac{9 - 1 \cdot x^{2}}{9 x^{2} x + 9 x^{2} \cdot - 3}$

Step 4.2

Rewrite $- 1 x^{2}$ as $- x^{2}$ .

$\frac{9 - x^{2}}{9 x^{2} x + 9 x^{2} \cdot - 3}$

Step 4.3

Rewrite $9 - x^{2}$ in a factored form.

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Step 4.3.1

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

$\frac{3^{2} - x^{2}}{9 x^{2} x + 9 x^{2} \cdot - 3}$

Step 4.3.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3$ and $b = x$ .

$\frac{(3 + x) (3 - x)}{9 x^{2} x + 9 x^{2} \cdot - 3}$

Step 5

Simplify the denominator.

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Step 5.1

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 5.1.1

Move $x$ .

$\frac{(3 + x) (3 - x)}{9 (x \cdot x^{2}) + 9 x^{2} \cdot - 3}$

Step 5.1.2

Multiply $x$ by $x^{2}$ .

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Step 5.1.2.1

Raise $x$ to the power of $1$ .

$\frac{(3 + x) (3 - x)}{9 (x^{1} x^{2}) + 9 x^{2} \cdot - 3}$

Step 5.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{(3 + x) (3 - x)}{9 x^{1 + 2} + 9 x^{2} \cdot - 3}$

Step 5.1.3

Add $1$ and $2$ .

$\frac{(3 + x) (3 - x)}{9 x^{3} + 9 x^{2} \cdot - 3}$

Step 5.2

Multiply $- 3$ by $9$ .

$\frac{(3 + x) (3 - x)}{9 x^{3} - 27 x^{2}}$

Step 5.3

Factor $9 x^{2}$ out of $9 x^{3} - 27 x^{2}$ .

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Step 5.3.1

Factor $9 x^{2}$ out of $9 x^{3}$ .

$\frac{(3 + x) (3 - x)}{9 x^{2} (x) - 27 x^{2}}$

Step 5.3.2

Factor $9 x^{2}$ out of $- 27 x^{2}$ .

$\frac{(3 + x) (3 - x)}{9 x^{2} (x) + 9 x^{2} (- 3)}$

Step 5.3.3

Factor $9 x^{2}$ out of $9 x^{2} (x) + 9 x^{2} (- 3)$ .

$\frac{(3 + x) (3 - x)}{9 x^{2} (x - 3)}$

Step 6

Cancel the common factor of $3 - x$ and $x - 3$ .

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Step 6.1

Rewrite $3$ as $- 1 (- 3)$ .

$\frac{(3 + x) (- 1 (- 3) - x)}{9 x^{2} (x - 3)}$

Step 6.2

Factor $- 1$ out of $- x$ .

$\frac{(3 + x) (- 1 (- 3) - (x))}{9 x^{2} (x - 3)}$

Step 6.3

Factor $- 1$ out of $- 1 (- 3) - (x)$ .

$\frac{(3 + x) (- 1 (- 3 + x))}{9 x^{2} (x - 3)}$

Step 6.4

Reorder terms.

$\frac{(3 + x) (- 1 (x - 3))}{9 x^{2} (x - 3)}$

Step 6.5

Cancel the common factor.

$\frac{(3 + x) (- 1 (x - 3))}{9 x^{2} (x - 3)}$

Step 6.6

Rewrite the expression.

$\frac{(3 + x) \cdot (- 1)}{9 x^{2}}$

Step 7

Move $- 1$ to the left of $3 + x$ .

$\frac{- 1 \cdot (3 + x)}{9 x^{2}}$

Step 8

Move the negative in front of the fraction.

$- \frac{3 + x}{9 x^{2}}$

Step 9

Split the fraction $\frac{3 + x}{9 x^{2}}$ into two fractions.

$- (\frac{3}{9 x^{2}} + \frac{x}{9 x^{2}})$

Step 10

Cancel the common factor of $3$ and $9$ .

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Step 10.1

Factor $3$ out of $3$ .

$- (\frac{3 (1)}{9 x^{2}} + \frac{x}{9 x^{2}})$

Step 10.2

Cancel the common factors.

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Step 10.2.1

Factor $3$ out of $9 x^{2}$ .

$- (\frac{3 (1)}{3 (3 x^{2})} + \frac{x}{9 x^{2}})$

Step 10.2.2

Cancel the common factor.

$- (\frac{3 \cdot 1}{3 (3 x^{2})} + \frac{x}{9 x^{2}})$

Step 10.2.3

Rewrite the expression.

$- (\frac{1}{3 x^{2}} + \frac{x}{9 x^{2}})$

Step 11

Cancel the common factor of $x$ and $x^{2}$ .

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Step 11.1

Raise $x$ to the power of $1$ .

$- (\frac{1}{3 x^{2}} + \frac{x^{1}}{9 x^{2}})$

Step 11.2

Factor $x$ out of $x^{1}$ .

$- (\frac{1}{3 x^{2}} + \frac{x \cdot 1}{9 x^{2}})$

Step 11.3

Cancel the common factors.

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Step 11.3.1

Factor $x$ out of $9 x^{2}$ .

$- (\frac{1}{3 x^{2}} + \frac{x \cdot 1}{x (9 x)})$

Step 11.3.2

Cancel the common factor.

$- (\frac{1}{3 x^{2}} + \frac{x \cdot 1}{x (9 x)})$

Step 11.3.3

Rewrite the expression.

$- (\frac{1}{3 x^{2}} + \frac{1}{9 x})$