Algebra Examples

$f (x) = \frac{x^{2} - 19 x + 88}{x^{2} - 121}$

Step 1

Factor $x^{2} - 19 x + 88$ using the AC method.

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Step 1.1

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 $88$ and whose sum is $- 19$ .

$- 11, - 8$

Step 1.2

Write the factored form using these integers.

$f (x) = \frac{(x - 11) (x - 8)}{x^{2} - 121}$

Step 2

Rewrite $121$ as $11^{2}$ .

$f (x) = \frac{(x - 11) (x - 8)}{x^{2} - 11^{2}}$

Step 3

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 $b = 11$ .

$f (x) = \frac{(x - 11) (x - 8)}{(x + 11) (x - 11)}$

Step 4

Reduce the expression $\frac{(x - 11) (x - 8)}{(x + 11) (x - 11)}$ by cancelling the common factors.

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Step 4.1

Cancel the common factor.

$f (x) = \frac{(x - 11) (x - 8)}{(x + 11) (x - 11)}$

Step 4.2

Rewrite the expression.

$f (x) = \frac{x - 8}{x + 11}$