Algebra Examples

$\frac{x^{2} - 25}{x^{2} + 8 x + 15} \cdot \frac{x^{2} + 3 x}{x - 5}$

Step 1

Simplify the numerator.

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Step 1.1

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

$\frac{x^{2} - 5^{2}}{x^{2} + 8 x + 15} \cdot \frac{x^{2} + 3 x}{x - 5}$

Step 1.2

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 $b = 5$ .

$\frac{(x + 5) (x - 5)}{x^{2} + 8 x + 15} \cdot \frac{x^{2} + 3 x}{x - 5}$

Step 2

Factor $x^{2} + 8 x + 15$ 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 $15$ and whose sum is $8$ .

$3, 5$

Step 2.2

Write the factored form using these integers.

$\frac{(x + 5) (x - 5)}{(x + 3) (x + 5)} \cdot \frac{x^{2} + 3 x}{x - 5}$

Step 3

Simplify terms.

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Step 3.1

Factor $x$ out of $x^{2} + 3 x$ .

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Step 3.1.1

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

$\frac{(x + 5) (x - 5)}{(x + 3) (x + 5)} \cdot \frac{x \cdot x + 3 x}{x - 5}$

Step 3.1.2

Factor $x$ out of $3 x$ .

$\frac{(x + 5) (x - 5)}{(x + 3) (x + 5)} \cdot \frac{x \cdot x + x \cdot 3}{x - 5}$

Step 3.1.3

Factor $x$ out of $x \cdot x + x \cdot 3$ .

$\frac{(x + 5) (x - 5)}{(x + 3) (x + 5)} \cdot \frac{x (x + 3)}{x - 5}$

Step 3.2

Cancel the common factor of $x - 5$ .

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Step 3.2.1

Factor $x - 5$ out of $(x + 5) (x - 5)$ .

$\frac{(x - 5) (x + 5)}{(x + 3) (x + 5)} \cdot \frac{x (x + 3)}{x - 5}$

Step 3.2.2

Cancel the common factor.

$\frac{(x - 5) (x + 5)}{(x + 3) (x + 5)} \cdot \frac{x (x + 3)}{x - 5}$

Step 3.2.3

Rewrite the expression.

$\frac{x + 5}{(x + 3) (x + 5)} \cdot (x (x + 3))$

Step 3.3

Cancel the common factor of $x + 3$ .

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Step 3.3.1

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

$\frac{x + 5}{(x + 3) (x + 5)} \cdot ((x + 3) x)$

Step 3.3.2

Cancel the common factor.

$\frac{x + 5}{(x + 3) (x + 5)} \cdot ((x + 3) x)$

Step 3.3.3

Rewrite the expression.

$\frac{x + 5}{x + 5} \cdot x$

Step 3.4

Combine $\frac{x + 5}{x + 5}$ and $x$ .

$\frac{(x + 5) x}{x + 5}$

Step 3.5

Cancel the common factor of $x + 5$ .

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Step 3.5.1

Cancel the common factor.

$\frac{(x + 5) x}{x + 5}$

Step 3.5.2

Divide $x$ by $1$ .

$x$