Algebra Examples

$\frac{10 x - 8}{x + 2} \div \frac{25 x^{2} - 16}{2 x - 4}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{10 x - 8}{x + 2} \cdot \frac{2 x - 4}{25 x^{2} - 16}$

Step 2

Factor $2$ out of $10 x - 8$ .

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Step 2.1

Factor $2$ out of $10 x$ .

$\frac{2 (5 x) - 8}{x + 2} \cdot \frac{2 x - 4}{25 x^{2} - 16}$

Step 2.2

Factor $2$ out of $- 8$ .

$\frac{2 (5 x) + 2 (- 4)}{x + 2} \cdot \frac{2 x - 4}{25 x^{2} - 16}$

Step 2.3

Factor $2$ out of $2 (5 x) + 2 (- 4)$ .

$\frac{2 (5 x - 4)}{x + 2} \cdot \frac{2 x - 4}{25 x^{2} - 16}$

Step 3

Factor $2$ out of $2 x - 4$ .

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Step 3.1

Factor $2$ out of $2 x$ .

$\frac{2 (5 x - 4)}{x + 2} \cdot \frac{2 (x) - 4}{25 x^{2} - 16}$

Step 3.2

Factor $2$ out of $- 4$ .

$\frac{2 (5 x - 4)}{x + 2} \cdot \frac{2 x + 2 \cdot - 2}{25 x^{2} - 16}$

Step 3.3

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

$\frac{2 (5 x - 4)}{x + 2} \cdot \frac{2 (x - 2)}{25 x^{2} - 16}$

Step 4

Simplify the denominator.

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Step 4.1

Rewrite $25 x^{2}$ as ${(5 x)}^{2}$ .

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

Step 4.2

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

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

Step 4.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 5 x$ and $b = 4$ .

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

Step 5

Cancel the common factor of $5 x - 4$ .

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Step 5.1

Factor $5 x - 4$ out of $2 (5 x - 4)$ .

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

Step 5.2

Factor $5 x - 4$ out of $(5 x + 4) (5 x - 4)$ .

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

Step 5.3

Cancel the common factor.

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

Step 5.4

Rewrite the expression.

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

Step 6

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

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

Step 7

Multiply $2$ by $2$ .

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

Step 8

Apply the distributive property.

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

Step 9

Multiply $4$ by $- 2$ .

$\frac{4 x - 8}{(x + 2) (5 x + 4)}$

Step 10

Split the fraction $\frac{4 x - 8}{(x + 2) (5 x + 4)}$ into two fractions.

$\frac{4 x}{(x + 2) (5 x + 4)} + \frac{- 8}{(x + 2) (5 x + 4)}$

Step 11

Move the negative in front of the fraction.

$\frac{4 x}{(x + 2) (5 x + 4)} - \frac{8}{(x + 2) (5 x + 4)}$