Algebra Examples

$\frac{28}{x^{2} - 9} + \frac{2 x}{x - 3} + \frac{6}{x + 3} = 0$

Step 1

To write $\frac{28}{x^{2} - 9}$ as a fraction with a common denominator, multiply by $\frac{x - 3}{x - 3}$ .

$\frac{28}{x^{2} - 9} \cdot \frac{x - 3}{x - 3} + \frac{2 x}{x - 3} + \frac{6}{x + 3} = 0$

Step 2

To write $\frac{2 x}{x - 3}$ as a fraction with a common denominator, multiply by $\frac{x^{2} - 9}{x^{2} - 9}$ .

$\frac{28}{x^{2} - 9} \cdot \frac{x - 3}{x - 3} + \frac{2 x}{x - 3} \cdot \frac{x^{2} - 9}{x^{2} - 9} + \frac{6}{x + 3} = 0$

Step 3

Write each expression with a common denominator of $(x^{2} - 9) (x - 3)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{28}{x^{2} - 9}$ by $\frac{x - 3}{x - 3}$ .

$\frac{28 (x - 3)}{(x^{2} - 9) (x - 3)} + \frac{2 x}{x - 3} \cdot \frac{x^{2} - 9}{x^{2} - 9} + \frac{6}{x + 3} = 0$

Step 3.2

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

$\frac{28 (x - 3)}{(x^{2} - 9) (x - 3)} + \frac{2 x (x^{2} - 9)}{(x - 3) (x^{2} - 9)} + \frac{6}{x + 3} = 0$

Step 3.3

Reorder the factors of $(x^{2} - 9) (x - 3)$ .

$\frac{28 (x - 3)}{(x - 3) (x^{2} - 9)} + \frac{2 x (x^{2} - 9)}{(x - 3) (x^{2} - 9)} + \frac{6}{x + 3} = 0$

Step 4

Combine the numerators over the common denominator.

$\frac{28 (x - 3) + 2 x (x^{2} - 9)}{(x - 3) (x^{2} - 9)} + \frac{6}{x + 3} = 0$

Step 5

To write $\frac{28 (x - 3) + 2 x (x^{2} - 9)}{(x - 3) (x^{2} - 9)}$ as a fraction with a common denominator, multiply by $\frac{x + 3}{x + 3}$ .

$\frac{28 (x - 3) + 2 x (x^{2} - 9)}{(x - 3) (x^{2} - 9)} \cdot \frac{x + 3}{x + 3} + \frac{6}{x + 3} = 0$

Step 6

To write $\frac{6}{x + 3}$ as a fraction with a common denominator, multiply by $\frac{(x - 3) (x^{2} - 9)}{(x - 3) (x^{2} - 9)}$ .

$\frac{28 (x - 3) + 2 x (x^{2} - 9)}{(x - 3) (x^{2} - 9)} \cdot \frac{x + 3}{x + 3} + \frac{6}{x + 3} \cdot \frac{(x - 3) (x^{2} - 9)}{(x - 3) (x^{2} - 9)} = 0$

Step 7

Write each expression with a common denominator of $(x - 3) (x^{2} - 9) (x + 3)$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.1

Multiply $\frac{28 (x - 3) + 2 x (x^{2} - 9)}{(x - 3) (x^{2} - 9)}$ by $\frac{x + 3}{x + 3}$ .

$\frac{(28 (x - 3) + 2 x (x^{2} - 9)) (x + 3)}{(x - 3) (x^{2} - 9) (x + 3)} + \frac{6}{x + 3} \cdot \frac{(x - 3) (x^{2} - 9)}{(x - 3) (x^{2} - 9)} = 0$

Step 7.2

Multiply $\frac{6}{x + 3}$ by $\frac{(x - 3) (x^{2} - 9)}{(x - 3) (x^{2} - 9)}$ .

$\frac{(28 (x - 3) + 2 x (x^{2} - 9)) (x + 3)}{(x - 3) (x^{2} - 9) (x + 3)} + \frac{6 ((x - 3) (x^{2} - 9))}{(x + 3) ((x - 3) (x^{2} - 9))} = 0$

Step 7.3

Reorder the factors of $(x - 3) (x^{2} - 9) (x + 3)$ .

$\frac{(28 (x - 3) + 2 x (x^{2} - 9)) (x + 3)}{(x + 3) (x - 3) (x^{2} - 9)} + \frac{6 ((x - 3) (x^{2} - 9))}{(x + 3) ((x - 3) (x^{2} - 9))} = 0$

Step 7.4

Reorder the factors of $(x + 3) ((x - 3) (x^{2} - 9))$ .

$\frac{(28 (x - 3) + 2 x (x^{2} - 9)) (x + 3)}{(x + 3) (x - 3) (x^{2} - 9)} + \frac{6 ((x - 3) (x^{2} - 9))}{(x + 3) (x - 3) (x^{2} - 9)} = 0$

Step 8

Combine the numerators over the common denominator.

$\frac{(28 (x - 3) + 2 x (x^{2} - 9)) (x + 3) + 6 ((x - 3) (x^{2} - 9))}{(x + 3) (x - 3) (x^{2} - 9)} = 0$

Step 9

Rewrite $\frac{(28 (x - 3) + 2 x (x^{2} - 9)) (x + 3) + 6 ((x - 3) (x^{2} - 9))}{(x + 3) (x - 3) (x^{2} - 9)}$ 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.

$\frac{(28 x + 28 \cdot - 3 + 2 x (x^{2} - 9)) (x + 3) + 6 ((x - 3) (x^{2} - 9))}{(x + 3) (x - 3) (x^{2} - 9)} = 0$

Step 9.1.2

Multiply $28$ by $- 3$ .

$\frac{(28 x - 84 + 2 x (x^{2} - 9)) (x + 3) + 6 ((x - 3) (x^{2} - 9))}{(x + 3) (x - 3) (x^{2} - 9)} = 0$

Step 9.1.3

Apply the distributive property.

$\frac{(28 x - 84 + 2 x \cdot x^{2} + 2 x \cdot - 9) (x + 3) + 6 ((x - 3) (x^{2} - 9))}{(x + 3) (x - 3) (x^{2} - 9)} = 0$

Step 9.1.4

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

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Step 9.1.4.1

Move $x^{2}$ .

$\frac{(28 x - 84 + 2 (x^{2} x) + 2 x \cdot - 9) (x + 3) + 6 ((x - 3) (x^{2} - 9))}{(x + 3) (x - 3) (x^{2} - 9)} = 0$

Step 9.1.4.2

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

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Step 9.1.4.2.1

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

$\frac{(28 x - 84 + 2 (x^{2} x) + 2 x \cdot - 9) (x + 3) + 6 ((x - 3) (x^{2} - 9))}{(x + 3) (x - 3) (x^{2} - 9)} = 0$

Step 9.1.4.2.2

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

$\frac{(28 x - 84 + 2 x^{2 + 1} + 2 x \cdot - 9) (x + 3) + 6 ((x - 3) (x^{2} - 9))}{(x + 3) (x - 3) (x^{2} - 9)} = 0$

Step 9.1.4.3

Add $2$ and $1$ .

$\frac{(28 x - 84 + 2 x^{3} + 2 x \cdot - 9) (x + 3) + 6 ((x - 3) (x^{2} - 9))}{(x + 3) (x - 3) (x^{2} - 9)} = 0$

Step 9.1.5

Multiply $- 9$ by $2$ .

$\frac{(28 x - 84 + 2 x^{3} - 18 x) (x + 3) + 6 ((x - 3) (x^{2} - 9))}{(x + 3) (x - 3) (x^{2} - 9)} = 0$

Step 9.2

Subtract $18 x$ from $28 x$ .

$\frac{(10 x - 84 + 2 x^{3}) (x + 3) + 6 ((x - 3) (x^{2} - 9))}{(x + 3) (x - 3) (x^{2} - 9)} = 0$

Step 9.3

Expand $(10 x - 84 + 2 x^{3}) (x + 3)$ by multiplying each term in the first expression by each term in the second expression.

$\frac{10 x \cdot x + 10 x \cdot 3 - 84 x - 84 \cdot 3 + 2 x^{3} x + 2 x^{3} \cdot 3 + 6 ((x - 3) (x^{2} - 9))}{(x + 3) (x - 3) (x^{2} - 9)} = 0$

Step 9.4

Simplify each term.

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Step 9.4.1

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

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Step 9.4.1.1

Move $x$ .

$\frac{10 (x \cdot x) + 10 x \cdot 3 - 84 x - 84 \cdot 3 + 2 x^{3} x + 2 x^{3} \cdot 3 + 6 ((x - 3) (x^{2} - 9))}{(x + 3) (x - 3) (x^{2} - 9)} = 0$

Step 9.4.1.2

Multiply $x$ by $x$ .

$\frac{10 x^{2} + 10 x \cdot 3 - 84 x - 84 \cdot 3 + 2 x^{3} x + 2 x^{3} \cdot 3 + 6 ((x - 3) (x^{2} - 9))}{(x + 3) (x - 3) (x^{2} - 9)} = 0$

Step 9.4.2

Multiply $3$ by $10$ .

$\frac{10 x^{2} + 30 x - 84 x - 84 \cdot 3 + 2 x^{3} x + 2 x^{3} \cdot 3 + 6 ((x - 3) (x^{2} - 9))}{(x + 3) (x - 3) (x^{2} - 9)} = 0$

Step 9.4.3

Multiply $- 84$ by $3$ .

$\frac{10 x^{2} + 30 x - 84 x - 252 + 2 x^{3} x + 2 x^{3} \cdot 3 + 6 ((x - 3) (x^{2} - 9))}{(x + 3) (x - 3) (x^{2} - 9)} = 0$

Step 9.4.4

Multiply $x^{3}$ by $x$ by adding the exponents.

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Step 9.4.4.1

Move $x$ .

$\frac{10 x^{2} + 30 x - 84 x - 252 + 2 (x \cdot x^{3}) + 2 x^{3} \cdot 3 + 6 ((x - 3) (x^{2} - 9))}{(x + 3) (x - 3) (x^{2} - 9)} = 0$

Step 9.4.4.2

Multiply $x$ by $x^{3}$ .

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Step 9.4.4.2.1

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

$\frac{10 x^{2} + 30 x - 84 x - 252 + 2 (x \cdot x^{3}) + 2 x^{3} \cdot 3 + 6 ((x - 3) (x^{2} - 9))}{(x + 3) (x - 3) (x^{2} - 9)} = 0$

Step 9.4.4.2.2

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

$\frac{10 x^{2} + 30 x - 84 x - 252 + 2 x^{1 + 3} + 2 x^{3} \cdot 3 + 6 ((x - 3) (x^{2} - 9))}{(x + 3) (x - 3) (x^{2} - 9)} = 0$

Step 9.4.4.3

Add $1$ and $3$ .

$\frac{10 x^{2} + 30 x - 84 x - 252 + 2 x^{4} + 2 x^{3} \cdot 3 + 6 ((x - 3) (x^{2} - 9))}{(x + 3) (x - 3) (x^{2} - 9)} = 0$

Step 9.4.5

Multiply $3$ by $2$ .

$\frac{10 x^{2} + 30 x - 84 x - 252 + 2 x^{4} + 6 x^{3} + 6 ((x - 3) (x^{2} - 9))}{(x + 3) (x - 3) (x^{2} - 9)} = 0$

Step 9.5

Subtract $84 x$ from $30 x$ .

$\frac{10 x^{2} - 54 x - 252 + 2 x^{4} + 6 x^{3} + 6 ((x - 3) (x^{2} - 9))}{(x + 3) (x - 3) (x^{2} - 9)} = 0$

Step 9.6

Expand $(x - 3) (x^{2} - 9)$ using the FOIL Method.

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Step 9.6.1

Apply the distributive property.

$\frac{10 x^{2} - 54 x - 252 + 2 x^{4} + 6 x^{3} + 6 (x (x^{2} - 9) - 3 (x^{2} - 9))}{(x + 3) (x - 3) (x^{2} - 9)} = 0$

Step 9.6.2

Apply the distributive property.

$\frac{10 x^{2} - 54 x - 252 + 2 x^{4} + 6 x^{3} + 6 (x \cdot x^{2} + x \cdot - 9 - 3 (x^{2} - 9))}{(x + 3) (x - 3) (x^{2} - 9)} = 0$

Step 9.6.3

Apply the distributive property.

$\frac{10 x^{2} - 54 x - 252 + 2 x^{4} + 6 x^{3} + 6 (x \cdot x^{2} + x \cdot - 9 - 3 x^{2} - 3 \cdot - 9)}{(x + 3) (x - 3) (x^{2} - 9)} = 0$

Step 9.7

Simplify each term.

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Step 9.7.1

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

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Step 9.7.1.1

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

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Step 9.7.1.1.1

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

$\frac{10 x^{2} - 54 x - 252 + 2 x^{4} + 6 x^{3} + 6 (x \cdot x^{2} + x \cdot - 9 - 3 x^{2} - 3 \cdot - 9)}{(x + 3) (x - 3) (x^{2} - 9)} = 0$

Step 9.7.1.1.2

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

$\frac{10 x^{2} - 54 x - 252 + 2 x^{4} + 6 x^{3} + 6 (x^{1 + 2} + x \cdot - 9 - 3 x^{2} - 3 \cdot - 9)}{(x + 3) (x - 3) (x^{2} - 9)} = 0$

Step 9.7.1.2

Add $1$ and $2$ .

$\frac{10 x^{2} - 54 x - 252 + 2 x^{4} + 6 x^{3} + 6 (x^{3} + x \cdot - 9 - 3 x^{2} - 3 \cdot - 9)}{(x + 3) (x - 3) (x^{2} - 9)} = 0$

Step 9.7.2

Move $- 9$ to the left of $x$ .

$\frac{10 x^{2} - 54 x - 252 + 2 x^{4} + 6 x^{3} + 6 (x^{3} - 9 x - 3 x^{2} - 3 \cdot - 9)}{(x + 3) (x - 3) (x^{2} - 9)} = 0$

Step 9.7.3

Multiply $- 3$ by $- 9$ .

$\frac{10 x^{2} - 54 x - 252 + 2 x^{4} + 6 x^{3} + 6 (x^{3} - 9 x - 3 x^{2} + 27)}{(x + 3) (x - 3) (x^{2} - 9)} = 0$

Step 9.8

Apply the distributive property.

$\frac{10 x^{2} - 54 x - 252 + 2 x^{4} + 6 x^{3} + 6 x^{3} + 6 (- 9 x) + 6 (- 3 x^{2}) + 6 \cdot 27}{(x + 3) (x - 3) (x^{2} - 9)} = 0$

Step 9.9

Simplify.

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Step 9.9.1

Multiply $- 9$ by $6$ .

$\frac{10 x^{2} - 54 x - 252 + 2 x^{4} + 6 x^{3} + 6 x^{3} - 54 x + 6 (- 3 x^{2}) + 6 \cdot 27}{(x + 3) (x - 3) (x^{2} - 9)} = 0$

Step 9.9.2

Multiply $- 3$ by $6$ .

$\frac{10 x^{2} - 54 x - 252 + 2 x^{4} + 6 x^{3} + 6 x^{3} - 54 x - 18 x^{2} + 6 \cdot 27}{(x + 3) (x - 3) (x^{2} - 9)} = 0$

Step 9.9.3

Multiply $6$ by $27$ .

$\frac{10 x^{2} - 54 x - 252 + 2 x^{4} + 6 x^{3} + 6 x^{3} - 54 x - 18 x^{2} + 162}{(x + 3) (x - 3) (x^{2} - 9)} = 0$

Step 9.10

Subtract $18 x^{2}$ from $10 x^{2}$ .

$\frac{- 8 x^{2} - 54 x - 252 + 2 x^{4} + 6 x^{3} + 6 x^{3} - 54 x + 162}{(x + 3) (x - 3) (x^{2} - 9)} = 0$

Step 9.11

Subtract $54 x$ from $- 54 x$ .

$\frac{- 8 x^{2} - 252 + 2 x^{4} + 6 x^{3} + 6 x^{3} - 108 x + 162}{(x + 3) (x - 3) (x^{2} - 9)} = 0$

Step 9.12

Add $- 252$ and $162$ .

$\frac{- 8 x^{2} + 2 x^{4} + 6 x^{3} + 6 x^{3} - 108 x - 90}{(x + 3) (x - 3) (x^{2} - 9)} = 0$

Step 9.13

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

$\frac{- 8 x^{2} + 2 x^{4} + 12 x^{3} - 108 x - 90}{(x + 3) (x - 3) (x^{2} - 9)} = 0$

Step 9.14

Reorder terms.

$\frac{2 x^{4} + 12 x^{3} - 8 x^{2} - 108 x - 90}{(x + 3) (x - 3) (x^{2} - 9)} = 0$

Step 9.15

Factor $2$ out of $2 x^{4} + 12 x^{3} - 8 x^{2} - 108 x - 90$ .

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Step 9.15.1

Factor $2$ out of $2 x^{4}$ .

$\frac{2 (x^{4}) + 12 x^{3} - 8 x^{2} - 108 x - 90}{(x + 3) (x - 3) (x^{2} - 9)} = 0$

Step 9.15.2

Factor $2$ out of $12 x^{3}$ .

$\frac{2 (x^{4}) + 2 (6 x^{3}) - 8 x^{2} - 108 x - 90}{(x + 3) (x - 3) (x^{2} - 9)} = 0$

Step 9.15.3

Factor $2$ out of $- 8 x^{2}$ .

$\frac{2 (x^{4}) + 2 (6 x^{3}) + 2 (- 4 x^{2}) - 108 x - 90}{(x + 3) (x - 3) (x^{2} - 9)} = 0$

Step 9.15.4

Factor $2$ out of $- 108 x$ .

$\frac{2 (x^{4}) + 2 (6 x^{3}) + 2 (- 4 x^{2}) + 2 (- 54 x) - 90}{(x + 3) (x - 3) (x^{2} - 9)} = 0$

Step 9.15.5

Factor $2$ out of $- 90$ .

$\frac{2 x^{4} + 2 (6 x^{3}) + 2 (- 4 x^{2}) + 2 (- 54 x) + 2 \cdot - 45}{(x + 3) (x - 3) (x^{2} - 9)} = 0$

Step 9.15.6

Factor $2$ out of $2 x^{4} + 2 (6 x^{3})$ .

$\frac{2 (x^{4} + 6 x^{3}) + 2 (- 4 x^{2}) + 2 (- 54 x) + 2 \cdot - 45}{(x + 3) (x - 3) (x^{2} - 9)} = 0$

Step 9.15.7

Factor $2$ out of $2 (x^{4} + 6 x^{3}) + 2 (- 4 x^{2})$ .

$\frac{2 (x^{4} + 6 x^{3} - 4 x^{2}) + 2 (- 54 x) + 2 \cdot - 45}{(x + 3) (x - 3) (x^{2} - 9)} = 0$

Step 9.15.8

Factor $2$ out of $2 (x^{4} + 6 x^{3} - 4 x^{2}) + 2 (- 54 x)$ .

$\frac{2 (x^{4} + 6 x^{3} - 4 x^{2} - 54 x) + 2 \cdot - 45}{(x + 3) (x - 3) (x^{2} - 9)} = 0$

Step 9.15.9

Factor $2$ out of $2 (x^{4} + 6 x^{3} - 4 x^{2} - 54 x) + 2 \cdot - 45$ .

$\frac{2 (x^{4} + 6 x^{3} - 4 x^{2} - 54 x - 45)}{(x + 3) (x - 3) (x^{2} - 9)} = 0$

Step 9.16

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

$\frac{2 (x^{4} + 6 x^{3} - 4 x^{2} - 54 x - 45)}{(x + 3) (x - 3) (x^{2} - 3^{2})} = 0$

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 $b = 3$ .

$\frac{2 (x^{4} + 6 x^{3} - 4 x^{2} - 54 x - 45)}{(x + 3) (x - 3) ((x + 3) (x - 3))} = 0$

Step 9.17.2

Remove unnecessary parentheses.

$\frac{2 (x^{4} + 6 x^{3} - 4 x^{2} - 54 x - 45)}{(x + 3) (x - 3) (x + 3) (x - 3)} = 0$

Step 9.18

Combine exponents.

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Step 9.18.1

Raise $x + 3$ to the power of $1$ .

$\frac{2 (x^{4} + 6 x^{3} - 4 x^{2} - 54 x - 45)}{(x + 3) (x + 3) (x - 3) (x - 3)} = 0$

Step 9.18.2

Raise $x + 3$ to the power of $1$ .

$\frac{2 (x^{4} + 6 x^{3} - 4 x^{2} - 54 x - 45)}{(x + 3) (x + 3) (x - 3) (x - 3)} = 0$

Step 9.18.3

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

$\frac{2 (x^{4} + 6 x^{3} - 4 x^{2} - 54 x - 45)}{{(x + 3)}^{1 + 1} (x - 3) (x - 3)} = 0$

Step 9.18.4

Add $1$ and $1$ .

$\frac{2 (x^{4} + 6 x^{3} - 4 x^{2} - 54 x - 45)}{{(x + 3)}^{2} (x - 3) (x - 3)} = 0$

Step 9.18.5

Raise $x - 3$ to the power of $1$ .

$\frac{2 (x^{4} + 6 x^{3} - 4 x^{2} - 54 x - 45)}{{(x + 3)}^{2} ((x - 3) (x - 3))} = 0$

Step 9.18.6

Raise $x - 3$ to the power of $1$ .

$\frac{2 (x^{4} + 6 x^{3} - 4 x^{2} - 54 x - 45)}{{(x + 3)}^{2} ((x - 3) (x - 3))} = 0$

Step 9.18.7

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

$\frac{2 (x^{4} + 6 x^{3} - 4 x^{2} - 54 x - 45)}{{(x + 3)}^{2} {(x - 3)}^{1 + 1}} = 0$

Step 9.18.8

Add $1$ and $1$ .

$\frac{2 (x^{4} + 6 x^{3} - 4 x^{2} - 54 x - 45)}{{(x + 3)}^{2} {(x - 3)}^{2}} = 0$

Step 10

Set the numerator equal to zero.

$2 (x^{4} + 6 x^{3} - 4 x^{2} - 54 x - 45) = 0$

Step 11

Solve the equation for $x$ .

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Step 11.1

Divide each term in $2 (x^{4} + 6 x^{3} - 4 x^{2} - 54 x - 45) = 0$ by $2$ and simplify.

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Step 11.1.1

Divide each term in $2 (x^{4} + 6 x^{3} - 4 x^{2} - 54 x - 45) = 0$ by $2$ .

$\frac{2 (x^{4} + 6 x^{3} - 4 x^{2} - 54 x - 45)}{2} = \frac{0}{2}$

Step 11.1.2

Simplify the left side.

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Step 11.1.2.1

Cancel the common factor of $2$ .

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Step 11.1.2.1.1

Cancel the common factor.

$\frac{2 (x^{4} + 6 x^{3} - 4 x^{2} - 54 x - 45)}{2} = \frac{0}{2}$

Step 11.1.2.1.2

Divide $x^{4} + 6 x^{3} - 4 x^{2} - 54 x - 45$ by $1$ .

$x^{4} + 6 x^{3} - 4 x^{2} - 54 x - 45 = \frac{0}{2}$

Step 11.1.3

Simplify the right side.

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Step 11.1.3.1

Divide $0$ by $2$ .

$x^{4} + 6 x^{3} - 4 x^{2} - 54 x - 45 = 0$

Step 11.2

Factor the left side of the equation.

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Step 11.2.1

Regroup terms.

$6 x^{3} - 54 x + x^{4} - 4 x^{2} - 45 = 0$

Step 11.2.2

Factor $6 x$ out of $6 x^{3} - 54 x$ .

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Step 11.2.2.1

Factor $6 x$ out of $6 x^{3}$ .

$6 x (x^{2}) - 54 x + x^{4} - 4 x^{2} - 45 = 0$

Step 11.2.2.2

Factor $6 x$ out of $- 54 x$ .

$6 x (x^{2}) + 6 x (- 9) + x^{4} - 4 x^{2} - 45 = 0$

Step 11.2.2.3

Factor $6 x$ out of $6 x (x^{2}) + 6 x (- 9)$ .

$6 x (x^{2} - 9) + x^{4} - 4 x^{2} - 45 = 0$

Step 11.2.3

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

$6 x (x^{2} - 3^{2}) + x^{4} - 4 x^{2} - 45 = 0$

Step 11.2.4

Factor.

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Step 11.2.4.1

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 $b = 3$ .

$6 x ((x + 3) (x - 3)) + x^{4} - 4 x^{2} - 45 = 0$

Step 11.2.4.2

Remove unnecessary parentheses.

$6 x (x + 3) (x - 3) + x^{4} - 4 x^{2} - 45 = 0$

Step 11.2.5

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$6 x (x + 3) (x - 3) + {(x^{2})}^{2} - 4 x^{2} - 45 = 0$

Step 11.2.6

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$6 x (x + 3) (x - 3) + u^{2} - 4 u - 45 = 0$

Step 11.2.7

Factor $u^{2} - 4 u - 45$ using the AC method.

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Step 11.2.7.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 45$ and whose sum is $- 4$ .

$- 9, 5$

Step 11.2.7.2

Write the factored form using these integers.

$6 x (x + 3) (x - 3) + (u - 9) (u + 5) = 0$

Step 11.2.8

Replace all occurrences of $u$ with $x^{2}$ .

$6 x (x + 3) (x - 3) + (x^{2} - 9) (x^{2} + 5) = 0$

Step 11.2.9

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

$6 x (x + 3) (x - 3) + (x^{2} - 3^{2}) (x^{2} + 5) = 0$

Step 11.2.10

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 $b = 3$ .

$6 x (x + 3) (x - 3) + (x + 3) (x - 3) (x^{2} + 5) = 0$

Step 11.2.11

Factor $(x + 3) (x - 3)$ out of $6 x (x + 3) (x - 3) + (x + 3) (x - 3) (x^{2} + 5)$ .

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Step 11.2.11.1

Factor $(x + 3) (x - 3)$ out of $6 x (x + 3) (x - 3)$ .

$(x + 3) (x - 3) (6 x) + (x + 3) (x - 3) (x^{2} + 5) = 0$

Step 11.2.11.2

Factor $(x + 3) (x - 3)$ out of $(x + 3) (x - 3) (6 x) + (x + 3) (x - 3) (x^{2} + 5)$ .

$(x + 3) (x - 3) (6 x + x^{2} + 5) = 0$

Step 11.2.12

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

$(x + 3) (x - 3) (6 u + u^{2} + 5) = 0$

Step 11.2.13

Factor $6 u + u^{2} + 5$ using the AC method.

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Step 11.2.13.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $5$ and whose sum is $6$ .

$1, 5$

Step 11.2.13.2

Write the factored form using these integers.

$(x + 3) (x - 3) ((u + 1) (u + 5)) = 0$

Step 11.2.14

Factor.

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Step 11.2.14.1

Replace all occurrences of $u$ with $x$ .

$(x + 3) (x - 3) ((x + 1) (x + 5)) = 0$

Step 11.2.14.2

Remove unnecessary parentheses.

$(x + 3) (x - 3) (x + 1) (x + 5) = 0$

Step 11.3

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

$x + 3 = 0$

$x - 3 = 0$

$x + 1 = 0$

$x + 5 = 0$

Step 11.4

Set $x + 3$ equal to $0$ and solve for $x$ .

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Step 11.4.1

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

$x + 3 = 0$

Step 11.4.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 11.5

Set $x - 3$ equal to $0$ and solve for $x$ .

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Step 11.5.1

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

$x - 3 = 0$

Step 11.5.2

Add $3$ to both sides of the equation.

$x = 3$

Step 11.6

Set $x + 1$ equal to $0$ and solve for $x$ .

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Step 11.6.1

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 11.6.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 11.7

Set $x + 5$ equal to $0$ and solve for $x$ .

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Step 11.7.1

Set $x + 5$ equal to $0$ .

$x + 5 = 0$

Step 11.7.2

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 11.8

The final solution is all the values that make $(x + 3) (x - 3) (x + 1) (x + 5) = 0$ true.

$x = - 3, 3, - 1, - 5$

Step 12

Exclude the solutions that do not make $\frac{2 (x^{4} + 6 x^{3} - 4 x^{2} - 54 x - 45)}{{(x + 3)}^{2} {(x - 3)}^{2}} = 0$ true.

$x = - 1, - 5$