Pre-Algebra Examples

$\frac{x^{2} + 7 x + 12}{x^{2} - 9} \div \frac{x + 4}{x - 4}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} + 7 x + 12}{x^{2} - 9} \cdot \frac{x - 4}{x + 4}$

Step 2

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

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Step 2.1

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 2.2

Write the factored form using these integers.

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

Step 3

Simplify the denominator.

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Step 3.1

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

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

Step 3.2

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

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

Step 4

Cancel the common factor of $x + 4$ .

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Step 4.1

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

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

Step 4.2

Cancel the common factor.

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

Step 4.3

Rewrite the expression.

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

Step 5

Cancel the common factor of $x + 3$ .

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Step 5.1

Cancel the common factor.

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

Step 5.2

Rewrite the expression.

$\frac{1}{x - 3} (x - 4)$

Step 6

Multiply $\frac{1}{x - 3}$ by $x - 4$ .

$\frac{x - 4}{x - 3}$