Pre-Algebra Examples

$\frac{x}{x^{2} + 7 x + 12} + \frac{4}{x^{2} - 16}$

Step 1

Simplify each term.

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Step 1.1

Factor $x^{2} + 7 x + 12$ using the AC method.

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Step 1.1.1

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 $12$ and whose sum is $7$ .

$3, 4$

Step 1.1.2

Write the factored form using these integers.

$\frac{x}{(x + 3) (x + 4)} + \frac{4}{x^{2} - 16}$

Step 1.2

Simplify the denominator.

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Step 1.2.1

Rewrite $16$ as $4^{2}$ .

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

Step 1.2.2

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 $b = 4$ .

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

Step 2

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

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

Step 3

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

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

Step 4

Write each expression with a common denominator of $(x + 3) (x + 4) (x - 4)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Reorder the factors of $(x + 4) (x - 4) (x + 3)$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Multiply $x$ by $x$ .

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

Step 6.3

Move $- 4$ to the left of $x$ .

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

Step 6.4

Apply the distributive property.

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

Step 6.5

Multiply $4$ by $3$ .

$\frac{x^{2} - 4 x + 4 x + 12}{(x + 3) (x + 4) (x - 4)}$

Step 6.6

Add $- 4 x$ and $4 x$ .

$\frac{x^{2} + 0 + 12}{(x + 3) (x + 4) (x - 4)}$

Step 6.7

Add $x^{2}$ and $0$ .

$\frac{x^{2} + 12}{(x + 3) (x + 4) (x - 4)}$