Algebra Examples

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

Step 1

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

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Step 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 1.2

Write the factored form using these integers.

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

Step 2

Simplify the denominator.

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Step 2.1

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

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

Step 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 $b = 2$ .

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

Step 3

Simplify the denominator.

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Step 3.1

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

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

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 $b = 4$ .

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

Step 4

Simplify terms.

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Step 4.1

Cancel the common factor of $x + 4$ .

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Step 4.1.1

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

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

Step 4.1.2

Cancel the common factor.

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

Step 4.1.3

Rewrite the expression.

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

Step 4.2

Cancel the common factor of $x + 2$ .

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Step 4.2.1

Cancel the common factor.

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

Step 4.2.2

Rewrite the expression.

$\frac{x + 3}{x - 2} \cdot \frac{1}{x - 4}$

Step 4.3

Multiply $\frac{x + 3}{x - 2}$ by $\frac{1}{x - 4}$ .

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