Algebra Examples

$\frac{x + 8}{x^{2} + 13 x + 40} \cdot \frac{x^{2} - 25}{9 x^{2}}$

Step 1

Factor $x^{2} + 13 x + 40$ 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 $40$ and whose sum is $13$ .

$5, 8$

Step 1.2

Write the factored form using these integers.

$\frac{x + 8}{(x + 5) (x + 8)} \cdot \frac{x^{2} - 25}{9 x^{2}}$

Step 2

Simplify the numerator.

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Step 2.1

Rewrite $25$ as $5^{2}$ .

$\frac{x + 8}{(x + 5) (x + 8)} \cdot \frac{x^{2} - 5^{2}}{9 x^{2}}$

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 $b = 5$ .

$\frac{x + 8}{(x + 5) (x + 8)} \cdot \frac{(x + 5) (x - 5)}{9 x^{2}}$

Step 3

Simplify terms.

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Step 3.1

Cancel the common factor of $x + 5$ .

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Step 3.1.1

Cancel the common factor.

$\frac{x + 8}{(x + 5) (x + 8)} \cdot \frac{(x + 5) (x - 5)}{9 x^{2}}$

Step 3.1.2

Rewrite the expression.

$\frac{x + 8}{x + 8} \cdot \frac{x - 5}{9 x^{2}}$

Step 3.2

Multiply $\frac{x + 8}{x + 8}$ by $\frac{x - 5}{9 x^{2}}$ .

$\frac{(x + 8) (x - 5)}{(x + 8) (9 x^{2})}$

Step 3.3

Cancel the common factor of $x + 8$ .

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Step 3.3.1

Cancel the common factor.

$\frac{(x + 8) (x - 5)}{(x + 8) (9 x^{2})}$

Step 3.3.2

Rewrite the expression.

$\frac{x - 5}{9 x^{2}}$