Algebra Examples

$\frac{x^{2} + 8 x}{(x^{2} + 11 x + 24)} \div \frac{(x - 3)}{(x^{2} - 9)}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} + 8 x}{x^{2} + 11 x + 24} \cdot \frac{x^{2} - 9}{x - 3}$

Step 2

Factor $x$ out of $x^{2} + 8 x$ .

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Step 2.1

Factor $x$ out of $x^{2}$ .

$\frac{x \cdot x + 8 x}{x^{2} + 11 x + 24} \cdot \frac{x^{2} - 9}{x - 3}$

Step 2.2

Factor $x$ out of $8 x$ .

$\frac{x \cdot x + x \cdot 8}{x^{2} + 11 x + 24} \cdot \frac{x^{2} - 9}{x - 3}$

Step 2.3

Factor $x$ out of $x \cdot x + x \cdot 8$ .

$\frac{x (x + 8)}{x^{2} + 11 x + 24} \cdot \frac{x^{2} - 9}{x - 3}$

Step 3

Factor $x^{2} + 11 x + 24$ using the AC method.

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Step 3.1

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 $24$ and whose sum is $11$ .

$3, 8$

Step 3.2

Write the factored form using these integers.

$\frac{x (x + 8)}{(x + 3) (x + 8)} \cdot \frac{x^{2} - 9}{x - 3}$

Step 4

Cancel the common factor of $x + 8$ .

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Step 4.1

Cancel the common factor.

$\frac{x (x + 8)}{(x + 3) (x + 8)} \cdot \frac{x^{2} - 9}{x - 3}$

Step 4.2

Rewrite the expression.

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

Step 5

Simplify the numerator.

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Step 5.1

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

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

Step 5.2

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 $b = 3$ .

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

Step 6

Simplify terms.

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Step 6.1

Cancel the common factor of $x + 3$ .

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Step 6.1.1

Cancel the common factor.

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

Step 6.1.2

Rewrite the expression.

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

Step 6.2

Combine $x$ and $\frac{x - 3}{x - 3}$ .

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

Step 6.3

Cancel the common factor of $x - 3$ .

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Step 6.3.1

Cancel the common factor.

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

Step 6.3.2

Divide $x$ by $1$ .

$x$