Algebra Examples

$\frac{x^{2}}{x^{2} - 5 x - 36} - \frac{36}{x^{2} - 5 x - 36}$

Step 1

Combine the numerators over the common denominator.

$\frac{x^{2} - 36}{x^{2} - 5 x - 36}$

Step 2

Simplify the numerator.

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Step 2.1

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

$\frac{x^{2} - 6^{2}}{x^{2} - 5 x - 36}$

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 $b = 6$ .

$\frac{(x + 6) (x - 6)}{x^{2} - 5 x - 36}$

Step 3

Factor $x^{2} - 5 x - 36$ using the AC method.

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Step 3.1

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 $- 36$ and whose sum is $- 5$ .

$- 9, 4$

Step 3.2

Write the factored form using these integers.

$\frac{(x + 6) (x - 6)}{(x - 9) (x + 4)}$