Pre-Algebra Examples

$\frac{x^{2} - 2 x - 48}{x^{2} - 12 x + 36} \div \frac{x - 8}{x - 6}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} - 2 x - 48}{x^{2} - 12 x + 36} \cdot \frac{x - 6}{x - 8}$

Step 2

Factor $x^{2} - 2 x - 48$ 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 $- 48$ and whose sum is $- 2$ .

$- 8, 6$

Step 2.2

Write the factored form using these integers.

$\frac{(x - 8) (x + 6)}{x^{2} - 12 x + 36} \cdot \frac{x - 6}{x - 8}$

Step 3

Factor using the perfect square rule.

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Step 3.1

Rewrite $36$ as $6^{2}$ .

$\frac{(x - 8) (x + 6)}{x^{2} - 12 x + 6^{2}} \cdot \frac{x - 6}{x - 8}$

Step 3.2

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

$12 x = 2 \cdot x \cdot 6$

Step 3.3

Rewrite the polynomial.

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

Step 3.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 6$ .

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

Step 4

Cancel the common factor of $x - 8$ .

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Step 4.1

Cancel the common factor.

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

Step 4.2

Rewrite the expression.

$\frac{x + 6}{{(x - 6)}^{2}} (x - 6)$

Step 5

Cancel the common factor of $x - 6$ .

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Step 5.1

Factor $x - 6$ out of ${(x - 6)}^{2}$ .

$\frac{x + 6}{(x - 6) (x - 6)} (x - 6)$

Step 5.2

Cancel the common factor.

$\frac{x + 6}{(x - 6) (x - 6)} (x - 6)$

Step 5.3

Rewrite the expression.

$\frac{x + 6}{x - 6}$