Algebra Examples

$\frac{x^{2} - 25}{x^{2} - 16} \cdot \frac{x^{2} - 4 x}{2 x + 10}$

Step 1

Simplify the numerator.

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Step 1.1

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

$\frac{x^{2} - 5^{2}}{x^{2} - 16} \cdot \frac{x^{2} - 4 x}{2 x + 10}$

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 $b = 5$ .

$\frac{(x + 5) (x - 5)}{x^{2} - 16} \cdot \frac{x^{2} - 4 x}{2 x + 10}$

Step 2

Simplify the denominator.

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Step 2.1

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

$\frac{(x + 5) (x - 5)}{x^{2} - 4^{2}} \cdot \frac{x^{2} - 4 x}{2 x + 10}$

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 $b = 4$ .

$\frac{(x + 5) (x - 5)}{(x + 4) (x - 4)} \cdot \frac{x^{2} - 4 x}{2 x + 10}$

Step 3

Simplify terms.

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Step 3.1

Factor $x$ out of $x^{2} - 4 x$ .

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Step 3.1.1

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

$\frac{(x + 5) (x - 5)}{(x + 4) (x - 4)} \cdot \frac{x \cdot x - 4 x}{2 x + 10}$

Step 3.1.2

Factor $x$ out of $- 4 x$ .

$\frac{(x + 5) (x - 5)}{(x + 4) (x - 4)} \cdot \frac{x \cdot x + x \cdot - 4}{2 x + 10}$

Step 3.1.3

Factor $x$ out of $x \cdot x + x \cdot - 4$ .

$\frac{(x + 5) (x - 5)}{(x + 4) (x - 4)} \cdot \frac{x (x - 4)}{2 x + 10}$

Step 3.2

Factor $2$ out of $2 x + 10$ .

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Step 3.2.1

Factor $2$ out of $2 x$ .

$\frac{(x + 5) (x - 5)}{(x + 4) (x - 4)} \cdot \frac{x (x - 4)}{2 (x) + 10}$

Step 3.2.2

Factor $2$ out of $10$ .

$\frac{(x + 5) (x - 5)}{(x + 4) (x - 4)} \cdot \frac{x (x - 4)}{2 x + 2 \cdot 5}$

Step 3.2.3

Factor $2$ out of $2 x + 2 \cdot 5$ .

$\frac{(x + 5) (x - 5)}{(x + 4) (x - 4)} \cdot \frac{x (x - 4)}{2 (x + 5)}$

Step 3.3

Cancel the common factor of $x + 5$ .

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Step 3.3.1

Factor $x + 5$ out of $2 (x + 5)$ .

$\frac{(x + 5) (x - 5)}{(x + 4) (x - 4)} \cdot \frac{x (x - 4)}{(x + 5) \cdot 2}$

Step 3.3.2

Cancel the common factor.

$\frac{(x + 5) (x - 5)}{(x + 4) (x - 4)} \cdot \frac{x (x - 4)}{(x + 5) \cdot 2}$

Step 3.3.3

Rewrite the expression.

$\frac{x - 5}{(x + 4) (x - 4)} \cdot \frac{x (x - 4)}{2}$

Step 3.4

Cancel the common factor of $x - 4$ .

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Step 3.4.1

Factor $x - 4$ out of $(x + 4) (x - 4)$ .

$\frac{x - 5}{(x - 4) (x + 4)} \cdot \frac{x (x - 4)}{2}$

Step 3.4.2

Factor $x - 4$ out of $x (x - 4)$ .

$\frac{x - 5}{(x - 4) (x + 4)} \cdot \frac{(x - 4) x}{2}$

Step 3.4.3

Cancel the common factor.

$\frac{x - 5}{(x - 4) (x + 4)} \cdot \frac{(x - 4) x}{2}$

Step 3.4.4

Rewrite the expression.

$\frac{x - 5}{x + 4} \cdot \frac{x}{2}$

Step 3.5

Multiply $\frac{x - 5}{x + 4}$ by $\frac{x}{2}$ .

$\frac{(x - 5) x}{(x + 4) \cdot 2}$

Step 3.6

Simplify the expression.

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Step 3.6.1

Move $2$ to the left of $x + 4$ .

$\frac{(x - 5) x}{2 \cdot (x + 4)}$

Step 3.6.2

Reorder factors in $\frac{(x - 5) x}{2 (x + 4)}$ .

$\frac{x (x - 5)}{2 (x + 4)}$