Algebra Examples

$\frac{2 x + 14}{x^{2} - 25} \cdot \frac{8 x + 40}{6 x + 42}$

Step 1

Factor $2$ out of $2 x + 14$ .

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Step 1.1

Factor $2$ out of $2 x$ .

$\frac{2 (x) + 14}{x^{2} - 25} \cdot \frac{8 x + 40}{6 x + 42}$

Step 1.2

Factor $2$ out of $14$ .

$\frac{2 x + 2 \cdot 7}{x^{2} - 25} \cdot \frac{8 x + 40}{6 x + 42}$

Step 1.3

Factor $2$ out of $2 x + 2 \cdot 7$ .

$\frac{2 (x + 7)}{x^{2} - 25} \cdot \frac{8 x + 40}{6 x + 42}$

Step 2

Simplify the denominator.

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Step 2.1

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

$\frac{2 (x + 7)}{x^{2} - 5^{2}} \cdot \frac{8 x + 40}{6 x + 42}$

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 $b = 5$ .

$\frac{2 (x + 7)}{(x + 5) (x - 5)} \cdot \frac{8 x + 40}{6 x + 42}$

Step 3

Simplify terms.

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Step 3.1

Factor $8$ out of $8 x + 40$ .

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Step 3.1.1

Factor $8$ out of $8 x$ .

$\frac{2 (x + 7)}{(x + 5) (x - 5)} \cdot \frac{8 (x) + 40}{6 x + 42}$

Step 3.1.2

Factor $8$ out of $40$ .

$\frac{2 (x + 7)}{(x + 5) (x - 5)} \cdot \frac{8 x + 8 \cdot 5}{6 x + 42}$

Step 3.1.3

Factor $8$ out of $8 x + 8 \cdot 5$ .

$\frac{2 (x + 7)}{(x + 5) (x - 5)} \cdot \frac{8 (x + 5)}{6 x + 42}$

Step 3.2

Factor $6$ out of $6 x + 42$ .

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Step 3.2.1

Factor $6$ out of $6 x$ .

$\frac{2 (x + 7)}{(x + 5) (x - 5)} \cdot \frac{8 (x + 5)}{6 (x) + 42}$

Step 3.2.2

Factor $6$ out of $42$ .

$\frac{2 (x + 7)}{(x + 5) (x - 5)} \cdot \frac{8 (x + 5)}{6 x + 6 \cdot 7}$

Step 3.2.3

Factor $6$ out of $6 x + 6 \cdot 7$ .

$\frac{2 (x + 7)}{(x + 5) (x - 5)} \cdot \frac{8 (x + 5)}{6 (x + 7)}$

Step 3.3

Cancel the common factor of $2 (x + 7)$ .

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Step 3.3.1

Factor $2 (x + 7)$ out of $6 (x + 7)$ .

$\frac{2 (x + 7)}{(x + 5) (x - 5)} \cdot \frac{8 (x + 5)}{2 (x + 7) (3)}$

Step 3.3.2

Cancel the common factor.

$\frac{2 (x + 7)}{(x + 5) (x - 5)} \cdot \frac{8 (x + 5)}{2 (x + 7) \cdot 3}$

Step 3.3.3

Rewrite the expression.

$\frac{1}{(x + 5) (x - 5)} \cdot \frac{8 (x + 5)}{3}$

Step 3.4

Cancel the common factor of $x + 5$ .

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Step 3.4.1

Factor $x + 5$ out of $8 (x + 5)$ .

$\frac{1}{(x + 5) (x - 5)} \cdot \frac{(x + 5) \cdot 8}{3}$

Step 3.4.2

Cancel the common factor.

$\frac{1}{(x + 5) (x - 5)} \cdot \frac{(x + 5) \cdot 8}{3}$

Step 3.4.3

Rewrite the expression.

$\frac{1}{x - 5} \cdot \frac{8}{3}$

Step 3.5

Multiply $\frac{1}{x - 5}$ by $\frac{8}{3}$ .

$\frac{8}{(x - 5) \cdot 3}$

Step 3.6

Move $3$ to the left of $x - 5$ .

$\frac{8}{3 (x - 5)}$