Algebra Examples

$\frac{x + 4}{x^{2} + 5 x + 6} \cdot \frac{x + 3}{x^{2} - 16}$

Step 1

Factor $x^{2} + 5 x + 6$ 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 $6$ and whose sum is $5$ .

$2, 3$

Step 1.2

Write the factored form using these integers.

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

Step 2

Simplify the denominator.

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Step 2.1

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

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

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 $b = 4$ .

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

Step 3

Simplify terms.

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Step 3.1

Cancel the common factor of $x + 4$ .

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Step 3.1.1

Cancel the common factor.

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

Step 3.1.2

Rewrite the expression.

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

Step 3.2

Cancel the common factor of $x + 3$ .

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Step 3.2.1

Factor $x + 3$ out of $(x + 2) (x + 3)$ .

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 3.3

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

$\frac{1}{(x + 2) (x - 4)}$