Precalculus Examples

$\frac{3}{x + 6} + \frac{6 x}{x^{2} - 36} - \frac{20}{x - 6}$

Step 1

To write $\frac{3}{x + 6}$ as a fraction with a common denominator, multiply by $\frac{x^{2} - 36}{x^{2} - 36}$ .

$\frac{3}{x + 6} \cdot \frac{x^{2} - 36}{x^{2} - 36} + \frac{6 x}{x^{2} - 36} - \frac{20}{x - 6}$

Step 2

To write $\frac{6 x}{x^{2} - 36}$ as a fraction with a common denominator, multiply by $\frac{x + 6}{x + 6}$ .

$\frac{3}{x + 6} \cdot \frac{x^{2} - 36}{x^{2} - 36} + \frac{6 x}{x^{2} - 36} \cdot \frac{x + 6}{x + 6} - \frac{20}{x - 6}$

Step 3

Write each expression with a common denominator of $(x + 6) (x^{2} - 36)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{3}{x + 6}$ by $\frac{x^{2} - 36}{x^{2} - 36}$ .

$\frac{3 (x^{2} - 36)}{(x + 6) (x^{2} - 36)} + \frac{6 x}{x^{2} - 36} \cdot \frac{x + 6}{x + 6} - \frac{20}{x - 6}$

Step 3.2

Multiply $\frac{6 x}{x^{2} - 36}$ by $\frac{x + 6}{x + 6}$ .

$\frac{3 (x^{2} - 36)}{(x + 6) (x^{2} - 36)} + \frac{6 x (x + 6)}{(x^{2} - 36) (x + 6)} - \frac{20}{x - 6}$

Step 3.3

Reorder the factors of $(x^{2} - 36) (x + 6)$ .

$\frac{3 (x^{2} - 36)}{(x + 6) (x^{2} - 36)} + \frac{6 x (x + 6)}{(x + 6) (x^{2} - 36)} - \frac{20}{x - 6}$

Step 4

Combine the numerators over the common denominator.

$\frac{3 (x^{2} - 36) + 6 x (x + 6)}{(x + 6) (x^{2} - 36)} - \frac{20}{x - 6}$

Step 5

To write $\frac{3 (x^{2} - 36) + 6 x (x + 6)}{(x + 6) (x^{2} - 36)}$ as a fraction with a common denominator, multiply by $\frac{x - 6}{x - 6}$ .

$\frac{3 (x^{2} - 36) + 6 x (x + 6)}{(x + 6) (x^{2} - 36)} \cdot \frac{x - 6}{x - 6} - \frac{20}{x - 6}$

Step 6

To write $- \frac{20}{x - 6}$ as a fraction with a common denominator, multiply by $\frac{(x + 6) (x^{2} - 36)}{(x + 6) (x^{2} - 36)}$ .

$\frac{3 (x^{2} - 36) + 6 x (x + 6)}{(x + 6) (x^{2} - 36)} \cdot \frac{x - 6}{x - 6} - \frac{20}{x - 6} \cdot \frac{(x + 6) (x^{2} - 36)}{(x + 6) (x^{2} - 36)}$

Step 7

Write each expression with a common denominator of $(x + 6) (x^{2} - 36) (x - 6)$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.1

Multiply $\frac{3 (x^{2} - 36) + 6 x (x + 6)}{(x + 6) (x^{2} - 36)}$ by $\frac{x - 6}{x - 6}$ .

$\frac{(3 (x^{2} - 36) + 6 x (x + 6)) (x - 6)}{(x + 6) (x^{2} - 36) (x - 6)} - \frac{20}{x - 6} \cdot \frac{(x + 6) (x^{2} - 36)}{(x + 6) (x^{2} - 36)}$

Step 7.2

Multiply $\frac{20}{x - 6}$ by $\frac{(x + 6) (x^{2} - 36)}{(x + 6) (x^{2} - 36)}$ .

$\frac{(3 (x^{2} - 36) + 6 x (x + 6)) (x - 6)}{(x + 6) (x^{2} - 36) (x - 6)} - \frac{20 ((x + 6) (x^{2} - 36))}{(x - 6) ((x + 6) (x^{2} - 36))}$

Step 7.3

Reorder the factors of $(x + 6) (x^{2} - 36) (x - 6)$ .

$\frac{(3 (x^{2} - 36) + 6 x (x + 6)) (x - 6)}{(x + 6) (x - 6) (x^{2} - 36)} - \frac{20 ((x + 6) (x^{2} - 36))}{(x - 6) ((x + 6) (x^{2} - 36))}$

Step 7.4

Reorder the factors of $(x - 6) ((x + 6) (x^{2} - 36))$ .

$\frac{(3 (x^{2} - 36) + 6 x (x + 6)) (x - 6)}{(x + 6) (x - 6) (x^{2} - 36)} - \frac{20 ((x + 6) (x^{2} - 36))}{(x + 6) (x - 6) (x^{2} - 36)}$

Step 8

Combine the numerators over the common denominator.

$\frac{(3 (x^{2} - 36) + 6 x (x + 6)) (x - 6) - 20 ((x + 6) (x^{2} - 36))}{(x + 6) (x - 6) (x^{2} - 36)}$

Step 9

Rewrite $\frac{(3 (x^{2} - 36) + 6 x (x + 6)) (x - 6) - 20 ((x + 6) (x^{2} - 36))}{(x + 6) (x - 6) (x^{2} - 36)}$ in a factored form.

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Step 9.1

Simplify each term.

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Step 9.1.1

Apply the distributive property.

$\frac{(3 x^{2} + 3 \cdot - 36 + 6 x (x + 6)) (x - 6) - 20 ((x + 6) (x^{2} - 36))}{(x + 6) (x - 6) (x^{2} - 36)}$

Step 9.1.2

Multiply $3$ by $- 36$ .

$\frac{(3 x^{2} - 108 + 6 x (x + 6)) (x - 6) - 20 ((x + 6) (x^{2} - 36))}{(x + 6) (x - 6) (x^{2} - 36)}$

Step 9.1.3

Apply the distributive property.

$\frac{(3 x^{2} - 108 + 6 x \cdot x + 6 x \cdot 6) (x - 6) - 20 ((x + 6) (x^{2} - 36))}{(x + 6) (x - 6) (x^{2} - 36)}$

Step 9.1.4

Multiply $x$ by $x$ by adding the exponents.

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Step 9.1.4.1

Move $x$ .

$\frac{(3 x^{2} - 108 + 6 (x \cdot x) + 6 x \cdot 6) (x - 6) - 20 ((x + 6) (x^{2} - 36))}{(x + 6) (x - 6) (x^{2} - 36)}$

Step 9.1.4.2

Multiply $x$ by $x$ .

$\frac{(3 x^{2} - 108 + 6 x^{2} + 6 x \cdot 6) (x - 6) - 20 ((x + 6) (x^{2} - 36))}{(x + 6) (x - 6) (x^{2} - 36)}$

Step 9.1.5

Multiply $6$ by $6$ .

$\frac{(3 x^{2} - 108 + 6 x^{2} + 36 x) (x - 6) - 20 ((x + 6) (x^{2} - 36))}{(x + 6) (x - 6) (x^{2} - 36)}$

Step 9.2

Add $3 x^{2}$ and $6 x^{2}$ .

$\frac{(9 x^{2} - 108 + 36 x) (x - 6) - 20 ((x + 6) (x^{2} - 36))}{(x + 6) (x - 6) (x^{2} - 36)}$

Step 9.3

Expand $(9 x^{2} - 108 + 36 x) (x - 6)$ by multiplying each term in the first expression by each term in the second expression.

$\frac{9 x^{2} x + 9 x^{2} \cdot - 6 - 108 x - 108 \cdot - 6 + 36 x \cdot x + 36 x \cdot - 6 - 20 ((x + 6) (x^{2} - 36))}{(x + 6) (x - 6) (x^{2} - 36)}$

Step 9.4

Simplify each term.

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Step 9.4.1

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

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Step 9.4.1.1

Move $x$ .

$\frac{9 (x \cdot x^{2}) + 9 x^{2} \cdot - 6 - 108 x - 108 \cdot - 6 + 36 x \cdot x + 36 x \cdot - 6 - 20 ((x + 6) (x^{2} - 36))}{(x + 6) (x - 6) (x^{2} - 36)}$

Step 9.4.1.2

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

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Step 9.4.1.2.1

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

$\frac{9 (x^{1} x^{2}) + 9 x^{2} \cdot - 6 - 108 x - 108 \cdot - 6 + 36 x \cdot x + 36 x \cdot - 6 - 20 ((x + 6) (x^{2} - 36))}{(x + 6) (x - 6) (x^{2} - 36)}$

Step 9.4.1.2.2

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

$\frac{9 x^{1 + 2} + 9 x^{2} \cdot - 6 - 108 x - 108 \cdot - 6 + 36 x \cdot x + 36 x \cdot - 6 - 20 ((x + 6) (x^{2} - 36))}{(x + 6) (x - 6) (x^{2} - 36)}$

Step 9.4.1.3

Add $1$ and $2$ .

$\frac{9 x^{3} + 9 x^{2} \cdot - 6 - 108 x - 108 \cdot - 6 + 36 x \cdot x + 36 x \cdot - 6 - 20 ((x + 6) (x^{2} - 36))}{(x + 6) (x - 6) (x^{2} - 36)}$

Step 9.4.2

Multiply $- 6$ by $9$ .

$\frac{9 x^{3} - 54 x^{2} - 108 x - 108 \cdot - 6 + 36 x \cdot x + 36 x \cdot - 6 - 20 ((x + 6) (x^{2} - 36))}{(x + 6) (x - 6) (x^{2} - 36)}$

Step 9.4.3

Multiply $- 108$ by $- 6$ .

$\frac{9 x^{3} - 54 x^{2} - 108 x + 648 + 36 x \cdot x + 36 x \cdot - 6 - 20 ((x + 6) (x^{2} - 36))}{(x + 6) (x - 6) (x^{2} - 36)}$

Step 9.4.4

Multiply $x$ by $x$ by adding the exponents.

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Step 9.4.4.1

Move $x$ .

$\frac{9 x^{3} - 54 x^{2} - 108 x + 648 + 36 (x \cdot x) + 36 x \cdot - 6 - 20 ((x + 6) (x^{2} - 36))}{(x + 6) (x - 6) (x^{2} - 36)}$

Step 9.4.4.2

Multiply $x$ by $x$ .

$\frac{9 x^{3} - 54 x^{2} - 108 x + 648 + 36 x^{2} + 36 x \cdot - 6 - 20 ((x + 6) (x^{2} - 36))}{(x + 6) (x - 6) (x^{2} - 36)}$

Step 9.4.5

Multiply $- 6$ by $36$ .

$\frac{9 x^{3} - 54 x^{2} - 108 x + 648 + 36 x^{2} - 216 x - 20 ((x + 6) (x^{2} - 36))}{(x + 6) (x - 6) (x^{2} - 36)}$

Step 9.5

Add $- 54 x^{2}$ and $36 x^{2}$ .

$\frac{9 x^{3} - 18 x^{2} - 108 x + 648 - 216 x - 20 ((x + 6) (x^{2} - 36))}{(x + 6) (x - 6) (x^{2} - 36)}$

Step 9.6

Subtract $216 x$ from $- 108 x$ .

$\frac{9 x^{3} - 18 x^{2} - 324 x + 648 - 20 ((x + 6) (x^{2} - 36))}{(x + 6) (x - 6) (x^{2} - 36)}$

Step 9.7

Expand $(x + 6) (x^{2} - 36)$ using the FOIL Method.

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Step 9.7.1

Apply the distributive property.

$\frac{9 x^{3} - 18 x^{2} - 324 x + 648 - 20 (x (x^{2} - 36) + 6 (x^{2} - 36))}{(x + 6) (x - 6) (x^{2} - 36)}$

Step 9.7.2

Apply the distributive property.

$\frac{9 x^{3} - 18 x^{2} - 324 x + 648 - 20 (x \cdot x^{2} + x \cdot - 36 + 6 (x^{2} - 36))}{(x + 6) (x - 6) (x^{2} - 36)}$

Step 9.7.3

Apply the distributive property.

$\frac{9 x^{3} - 18 x^{2} - 324 x + 648 - 20 (x \cdot x^{2} + x \cdot - 36 + 6 x^{2} + 6 \cdot - 36)}{(x + 6) (x - 6) (x^{2} - 36)}$

Step 9.8

Simplify each term.

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Step 9.8.1

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

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Step 9.8.1.1

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

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Step 9.8.1.1.1

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

$\frac{9 x^{3} - 18 x^{2} - 324 x + 648 - 20 (x^{1} x^{2} + x \cdot - 36 + 6 x^{2} + 6 \cdot - 36)}{(x + 6) (x - 6) (x^{2} - 36)}$

Step 9.8.1.1.2

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

$\frac{9 x^{3} - 18 x^{2} - 324 x + 648 - 20 (x^{1 + 2} + x \cdot - 36 + 6 x^{2} + 6 \cdot - 36)}{(x + 6) (x - 6) (x^{2} - 36)}$

Step 9.8.1.2

Add $1$ and $2$ .

$\frac{9 x^{3} - 18 x^{2} - 324 x + 648 - 20 (x^{3} + x \cdot - 36 + 6 x^{2} + 6 \cdot - 36)}{(x + 6) (x - 6) (x^{2} - 36)}$

Step 9.8.2

Move $- 36$ to the left of $x$ .

$\frac{9 x^{3} - 18 x^{2} - 324 x + 648 - 20 (x^{3} - 36 x + 6 x^{2} + 6 \cdot - 36)}{(x + 6) (x - 6) (x^{2} - 36)}$

Step 9.8.3

Multiply $6$ by $- 36$ .

$\frac{9 x^{3} - 18 x^{2} - 324 x + 648 - 20 (x^{3} - 36 x + 6 x^{2} - 216)}{(x + 6) (x - 6) (x^{2} - 36)}$

Step 9.9

Apply the distributive property.

$\frac{9 x^{3} - 18 x^{2} - 324 x + 648 - 20 x^{3} - 20 (- 36 x) - 20 (6 x^{2}) - 20 \cdot - 216}{(x + 6) (x - 6) (x^{2} - 36)}$

Step 9.10

Simplify.

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Step 9.10.1

Multiply $- 36$ by $- 20$ .

$\frac{9 x^{3} - 18 x^{2} - 324 x + 648 - 20 x^{3} + 720 x - 20 (6 x^{2}) - 20 \cdot - 216}{(x + 6) (x - 6) (x^{2} - 36)}$

Step 9.10.2

Multiply $6$ by $- 20$ .

$\frac{9 x^{3} - 18 x^{2} - 324 x + 648 - 20 x^{3} + 720 x - 120 x^{2} - 20 \cdot - 216}{(x + 6) (x - 6) (x^{2} - 36)}$

Step 9.10.3

Multiply $- 20$ by $- 216$ .

$\frac{9 x^{3} - 18 x^{2} - 324 x + 648 - 20 x^{3} + 720 x - 120 x^{2} + 4320}{(x + 6) (x - 6) (x^{2} - 36)}$

Step 9.11

Subtract $20 x^{3}$ from $9 x^{3}$ .

$\frac{- 11 x^{3} - 18 x^{2} - 324 x + 648 + 720 x - 120 x^{2} + 4320}{(x + 6) (x - 6) (x^{2} - 36)}$

Step 9.12

Subtract $120 x^{2}$ from $- 18 x^{2}$ .

$\frac{- 11 x^{3} - 138 x^{2} - 324 x + 648 + 720 x + 4320}{(x + 6) (x - 6) (x^{2} - 36)}$

Step 9.13

Add $- 324 x$ and $720 x$ .

$\frac{- 11 x^{3} - 138 x^{2} + 396 x + 648 + 4320}{(x + 6) (x - 6) (x^{2} - 36)}$

Step 9.14

Add $648$ and $4320$ .

$\frac{- 11 x^{3} - 138 x^{2} + 396 x + 4968}{(x + 6) (x - 6) (x^{2} - 36)}$

Step 9.15

Rewrite $- 11 x^{3} - 138 x^{2} + 396 x + 4968$ in a factored form.

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Step 9.15.1

Factor out the greatest common factor from each group.

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Step 9.15.1.1

Group the first two terms and the last two terms.

$\frac{(- 11 x^{3} - 138 x^{2}) + 396 x + 4968}{(x + 6) (x - 6) (x^{2} - 36)}$

Step 9.15.1.2

Factor out the greatest common factor (GCF) from each group.

$\frac{x^{2} (- 11 x - 138) - 36 (- 11 x - 138)}{(x + 6) (x - 6) (x^{2} - 36)}$

Step 9.15.2

Factor the polynomial by factoring out the greatest common factor, $- 11 x - 138$ .

$\frac{(- 11 x - 138) (x^{2} - 36)}{(x + 6) (x - 6) (x^{2} - 36)}$

Step 9.15.3

Rewrite $36$ as $6^{2}$ .

$\frac{(- 11 x - 138) (x^{2} - 6^{2})}{(x + 6) (x - 6) (x^{2} - 36)}$

Step 9.15.4

Factor.

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Step 9.15.4.1

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 $b = 6$ .

$\frac{(- 11 x - 138) ((x + 6) (x - 6))}{(x + 6) (x - 6) (x^{2} - 36)}$

Step 9.15.4.2

Remove unnecessary parentheses.

$\frac{(- 11 x - 138) (x + 6) (x - 6)}{(x + 6) (x - 6) (x^{2} - 36)}$

Step 9.16

Rewrite $36$ as $6^{2}$ .

$\frac{(- 11 x - 138) (x + 6) (x - 6)}{(x + 6) (x - 6) (x^{2} - 6^{2})}$

Step 9.17

Factor.

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Step 9.17.1

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 $b = 6$ .

$\frac{(- 11 x - 138) (x + 6) (x - 6)}{(x + 6) (x - 6) ((x + 6) (x - 6))}$

Step 9.17.2

Remove unnecessary parentheses.

$\frac{(- 11 x - 138) (x + 6) (x - 6)}{(x + 6) (x - 6) (x + 6) (x - 6)}$

Step 9.18

Combine exponents.

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Step 9.18.1

Raise $x + 6$ to the power of $1$ .

$\frac{(- 11 x - 138) (x + 6) (x - 6)}{{(x + 6)}^{1} (x + 6) (x - 6) (x - 6)}$

Step 9.18.2

Raise $x + 6$ to the power of $1$ .

$\frac{(- 11 x - 138) (x + 6) (x - 6)}{{(x + 6)}^{1} {(x + 6)}^{1} (x - 6) (x - 6)}$

Step 9.18.3

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

$\frac{(- 11 x - 138) (x + 6) (x - 6)}{{(x + 6)}^{1 + 1} (x - 6) (x - 6)}$

Step 9.18.4

Add $1$ and $1$ .

$\frac{(- 11 x - 138) (x + 6) (x - 6)}{{(x + 6)}^{2} (x - 6) (x - 6)}$

Step 9.18.5

Raise $x - 6$ to the power of $1$ .

$\frac{(- 11 x - 138) (x + 6) (x - 6)}{{(x + 6)}^{2} ({(x - 6)}^{1} (x - 6))}$

Step 9.18.6

Raise $x - 6$ to the power of $1$ .

$\frac{(- 11 x - 138) (x + 6) (x - 6)}{{(x + 6)}^{2} ({(x - 6)}^{1} {(x - 6)}^{1})}$

Step 9.18.7

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

$\frac{(- 11 x - 138) (x + 6) (x - 6)}{{(x + 6)}^{2} {(x - 6)}^{1 + 1}}$

Step 9.18.8

Add $1$ and $1$ .

$\frac{(- 11 x - 138) (x + 6) (x - 6)}{{(x + 6)}^{2} {(x - 6)}^{2}}$

Step 9.19

Reduce the expression $\frac{(- 11 x - 138) (x + 6) (x - 6)}{{(x + 6)}^{2} {(x - 6)}^{2}}$ by cancelling the common factors.

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Step 9.19.1

Factor $x + 6$ out of $(- 11 x - 138) (x + 6) (x - 6)$ .

$\frac{(x + 6) ((- 11 x - 138) (x - 6))}{{(x + 6)}^{2} {(x - 6)}^{2}}$

Step 9.19.2

Factor $x + 6$ out of ${(x + 6)}^{2} {(x - 6)}^{2}$ .

$\frac{(x + 6) ((- 11 x - 138) (x - 6))}{(x + 6) ((x + 6) {(x - 6)}^{2})}$

Step 9.19.3

Cancel the common factor.

$\frac{(x + 6) ((- 11 x - 138) (x - 6))}{(x + 6) ((x + 6) {(x - 6)}^{2})}$

Step 9.19.4

Rewrite the expression.

$\frac{(- 11 x - 138) (x - 6)}{(x + 6) {(x - 6)}^{2}}$

Step 10

Factor $- 1$ out of $- 11 x - 138$ .

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Step 10.1

Factor $- 1$ out of $- 11 x$ .

$\frac{(- (11 x) - 138) (x - 6)}{(x + 6) {(x - 6)}^{2}}$

Step 10.2

Rewrite $- 138$ as $- 1 (138)$ .

$\frac{(- (11 x) - 1 (138)) (x - 6)}{(x + 6) {(x - 6)}^{2}}$

Step 10.3

Factor $- 1$ out of $- (11 x) - 1 (138)$ .

$\frac{- (11 x + 138) (x - 6)}{(x + 6) {(x - 6)}^{2}}$

Step 11

Reduce the expression $\frac{- (11 x + 138) (x - 6)}{(x + 6) {(x - 6)}^{2}}$ by cancelling the common factors.

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Step 11.1

Factor $x - 6$ out of $- (11 x + 138) (x - 6)$ .

$\frac{(x - 6) (- (11 x + 138))}{(x + 6) {(x - 6)}^{2}}$

Step 11.2

Factor $x - 6$ out of $(x + 6) {(x - 6)}^{2}$ .

$\frac{(x - 6) (- (11 x + 138))}{(x - 6) ((x + 6) (x - 6))}$

Step 11.3

Cancel the common factor.

$\frac{(x - 6) (- (11 x + 138))}{(x - 6) ((x + 6) (x - 6))}$

Step 11.4

Rewrite the expression.

$\frac{- (11 x + 138)}{(x + 6) (x - 6)}$