Pre-Algebra Examples

$\frac{x^{2} - 25}{8 x - 8} \div \frac{x^{2} + 10 x + 25}{x^{2} + 4 x - 5}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} - 25}{8 x - 8} \cdot \frac{x^{2} + 4 x - 5}{x^{2} + 10 x + 25}$

Step 2

Simplify the numerator.

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Step 2.1

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

$\frac{x^{2} - 5^{2}}{8 x - 8} \cdot \frac{x^{2} + 4 x - 5}{x^{2} + 10 x + 25}$

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 $b = 5$ .

$\frac{(x + 5) (x - 5)}{8 x - 8} \cdot \frac{x^{2} + 4 x - 5}{x^{2} + 10 x + 25}$

Step 3

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

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Step 3.1

Factor $8$ out of $8 x$ .

$\frac{(x + 5) (x - 5)}{8 (x) - 8} \cdot \frac{x^{2} + 4 x - 5}{x^{2} + 10 x + 25}$

Step 3.2

Factor $8$ out of $- 8$ .

$\frac{(x + 5) (x - 5)}{8 (x) + 8 (- 1)} \cdot \frac{x^{2} + 4 x - 5}{x^{2} + 10 x + 25}$

Step 3.3

Factor $8$ out of $8 (x) + 8 (- 1)$ .

$\frac{(x + 5) (x - 5)}{8 (x - 1)} \cdot \frac{x^{2} + 4 x - 5}{x^{2} + 10 x + 25}$

Step 4

Factor $x^{2} + 4 x - 5$ using the AC method.

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Step 4.1

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 $- 5$ and whose sum is $4$ .

$- 1, 5$

Step 4.2

Write the factored form using these integers.

$\frac{(x + 5) (x - 5)}{8 (x - 1)} \cdot \frac{(x - 1) (x + 5)}{x^{2} + 10 x + 25}$

Step 5

Factor using the perfect square rule.

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Step 5.1

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

$\frac{(x + 5) (x - 5)}{8 (x - 1)} \cdot \frac{(x - 1) (x + 5)}{x^{2} + 10 x + 5^{2}}$

Step 5.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$10 x = 2 \cdot x \cdot 5$

Step 5.3

Rewrite the polynomial.

$\frac{(x + 5) (x - 5)}{8 (x - 1)} \cdot \frac{(x - 1) (x + 5)}{x^{2} + 2 \cdot x \cdot 5 + 5^{2}}$

Step 5.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 5$ .

$\frac{(x + 5) (x - 5)}{8 (x - 1)} \cdot \frac{(x - 1) (x + 5)}{{(x + 5)}^{2}}$

Step 6

Combine.

$\frac{(x + 5) (x - 5) ((x - 1) (x + 5))}{8 (x - 1) {(x + 5)}^{2}}$

Step 7

Cancel the common factor of $x + 5$ and ${(x + 5)}^{2}$ .

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Step 7.1

Factor $x + 5$ out of $(x + 5) (x - 5) ((x - 1) (x + 5))$ .

$\frac{(x + 5) ((x - 5) ((x - 1) (x + 5)))}{8 (x - 1) {(x + 5)}^{2}}$

Step 7.2

Cancel the common factors.

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Step 7.2.1

Factor $x + 5$ out of $8 (x - 1) {(x + 5)}^{2}$ .

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

Step 7.2.2

Cancel the common factor.

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

Step 7.2.3

Rewrite the expression.

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

Step 8

Cancel the common factor of $x - 1$ .

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Step 8.1

Cancel the common factor.

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

Step 8.2

Rewrite the expression.

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

Step 9

Cancel the common factor of $x + 5$ .

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Step 9.1

Cancel the common factor.

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

Step 9.2

Rewrite the expression.

$\frac{x - 5}{8}$

Step 10

Split the fraction $\frac{x - 5}{8}$ into two fractions.

$\frac{x}{8} + \frac{- 5}{8}$

Step 11

Move the negative in front of the fraction.

$\frac{x}{8} - \frac{5}{8}$