Precalculus Examples

$\frac{4 x + 12}{x^{2} + 6 x + 9} + \frac{5 x}{x^{2} - 9} + \frac{7}{x - 3}$

Step 1

Simplify each term.

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Step 1.1

Factor $4$ out of $4 x + 12$ .

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Step 1.1.1

Factor $4$ out of $4 x$ .

$\frac{4 (x) + 12}{x^{2} + 6 x + 9} + \frac{5 x}{x^{2} - 9} + \frac{7}{x - 3}$

Step 1.1.2

Factor $4$ out of $12$ .

$\frac{4 x + 4 \cdot 3}{x^{2} + 6 x + 9} + \frac{5 x}{x^{2} - 9} + \frac{7}{x - 3}$

Step 1.1.3

Factor $4$ out of $4 x + 4 \cdot 3$ .

$\frac{4 (x + 3)}{x^{2} + 6 x + 9} + \frac{5 x}{x^{2} - 9} + \frac{7}{x - 3}$

Step 1.2

Factor using the perfect square rule.

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Step 1.2.1

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

$\frac{4 (x + 3)}{x^{2} + 6 x + 3^{2}} + \frac{5 x}{x^{2} - 9} + \frac{7}{x - 3}$

Step 1.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$6 x = 2 \cdot x \cdot 3$

Step 1.2.3

Rewrite the polynomial.

$\frac{4 (x + 3)}{x^{2} + 2 \cdot x \cdot 3 + 3^{2}} + \frac{5 x}{x^{2} - 9} + \frac{7}{x - 3}$

Step 1.2.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 3$ .

$\frac{4 (x + 3)}{{(x + 3)}^{2}} + \frac{5 x}{x^{2} - 9} + \frac{7}{x - 3}$

Step 1.3

Cancel the common factor of $x + 3$ and ${(x + 3)}^{2}$ .

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Step 1.3.1

Factor $x + 3$ out of $4 (x + 3)$ .

$\frac{(x + 3) \cdot 4}{{(x + 3)}^{2}} + \frac{5 x}{x^{2} - 9} + \frac{7}{x - 3}$

Step 1.3.2

Cancel the common factors.

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Step 1.3.2.1

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

$\frac{(x + 3) \cdot 4}{(x + 3) (x + 3)} + \frac{5 x}{x^{2} - 9} + \frac{7}{x - 3}$

Step 1.3.2.2

Cancel the common factor.

$\frac{(x + 3) \cdot 4}{(x + 3) (x + 3)} + \frac{5 x}{x^{2} - 9} + \frac{7}{x - 3}$

Step 1.3.2.3

Rewrite the expression.

$\frac{4}{x + 3} + \frac{5 x}{x^{2} - 9} + \frac{7}{x - 3}$

Step 1.4

Simplify the denominator.

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Step 1.4.1

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

$\frac{4}{x + 3} + \frac{5 x}{x^{2} - 3^{2}} + \frac{7}{x - 3}$

Step 1.4.2

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 $b = 3$ .

$\frac{4}{x + 3} + \frac{5 x}{(x + 3) (x - 3)} + \frac{7}{x - 3}$

Step 2

To write $\frac{4}{x + 3}$ as a fraction with a common denominator, multiply by $\frac{x - 3}{x - 3}$ .

$\frac{4}{x + 3} \cdot \frac{x - 3}{x - 3} + \frac{5 x}{(x + 3) (x - 3)} + \frac{7}{x - 3}$

Step 3

Simplify terms.

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Step 3.1

Multiply $\frac{4}{x + 3}$ by $\frac{x - 3}{x - 3}$ .

$\frac{4 (x - 3)}{(x + 3) (x - 3)} + \frac{5 x}{(x + 3) (x - 3)} + \frac{7}{x - 3}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{4 (x - 3) + 5 x}{(x + 3) (x - 3)} + \frac{7}{x - 3}$

Step 4

Simplify the numerator.

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Step 4.1

Apply the distributive property.

$\frac{4 x + 4 \cdot - 3 + 5 x}{(x + 3) (x - 3)} + \frac{7}{x - 3}$

Step 4.2

Multiply $4$ by $- 3$ .

$\frac{4 x - 12 + 5 x}{(x + 3) (x - 3)} + \frac{7}{x - 3}$

Step 4.3

Add $4 x$ and $5 x$ .

$\frac{9 x - 12}{(x + 3) (x - 3)} + \frac{7}{x - 3}$

Step 4.4

Factor $3$ out of $9 x - 12$ .

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Step 4.4.1

Factor $3$ out of $9 x$ .

$\frac{3 (3 x) - 12}{(x + 3) (x - 3)} + \frac{7}{x - 3}$

Step 4.4.2

Factor $3$ out of $- 12$ .

$\frac{3 (3 x) + 3 (- 4)}{(x + 3) (x - 3)} + \frac{7}{x - 3}$

Step 4.4.3

Factor $3$ out of $3 (3 x) + 3 (- 4)$ .

$\frac{3 (3 x - 4)}{(x + 3) (x - 3)} + \frac{7}{x - 3}$

Step 5

To write $\frac{7}{x - 3}$ as a fraction with a common denominator, multiply by $\frac{x + 3}{x + 3}$ .

$\frac{3 (3 x - 4)}{(x + 3) (x - 3)} + \frac{7}{x - 3} \cdot \frac{x + 3}{x + 3}$

Step 6

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

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Step 6.1

Multiply $\frac{7}{x - 3}$ by $\frac{x + 3}{x + 3}$ .

$\frac{3 (3 x - 4)}{(x + 3) (x - 3)} + \frac{7 (x + 3)}{(x - 3) (x + 3)}$

Step 6.2

Reorder the factors of $(x - 3) (x + 3)$ .

$\frac{3 (3 x - 4)}{(x + 3) (x - 3)} + \frac{7 (x + 3)}{(x + 3) (x - 3)}$

Step 7

Combine the numerators over the common denominator.

$\frac{3 (3 x - 4) + 7 (x + 3)}{(x + 3) (x - 3)}$

Step 8

Simplify the numerator.

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Step 8.1

Apply the distributive property.

$\frac{3 (3 x) + 3 \cdot - 4 + 7 (x + 3)}{(x + 3) (x - 3)}$

Step 8.2

Multiply $3$ by $3$ .

$\frac{9 x + 3 \cdot - 4 + 7 (x + 3)}{(x + 3) (x - 3)}$

Step 8.3

Multiply $3$ by $- 4$ .

$\frac{9 x - 12 + 7 (x + 3)}{(x + 3) (x - 3)}$

Step 8.4

Apply the distributive property.

$\frac{9 x - 12 + 7 x + 7 \cdot 3}{(x + 3) (x - 3)}$

Step 8.5

Multiply $7$ by $3$ .

$\frac{9 x - 12 + 7 x + 21}{(x + 3) (x - 3)}$

Step 8.6

Add $9 x$ and $7 x$ .

$\frac{16 x - 12 + 21}{(x + 3) (x - 3)}$

Step 8.7

Add $- 12$ and $21$ .

$\frac{16 x + 9}{(x + 3) (x - 3)}$