Algebra Examples

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

Step 1

To divide by a fraction, multiply by its reciprocal.

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

Step 2

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

$- 3, - 2$

Step 2.2

Write the factored form using these integers.

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

Step 3

Simplify the denominator.

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Step 3.1

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

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

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 $b = 4$ .

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

Step 4

Cancel the common factor of $x + 4$ .

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Step 4.1

Cancel the common factor.

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

Step 4.2

Rewrite the expression.

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

Step 5

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

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

Step 6

Split the fraction $\frac{x + 3}{(x - 3) (x - 2) (x - 4)}$ into two fractions.

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