Finite Math Examples

$f (x) = \frac{x^{2} - 2 x - 63}{x^{2} - 64}$

Step 1

Factor $x^{2} - 2 x - 63$ 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 $- 63$ and whose sum is $- 2$ .

$- 9, 7$

Step 1.2

Write the factored form using these integers.

$f (x) = \frac{(x - 9) (x + 7)}{x^{2} - 64}$

Step 2

Rewrite $64$ as $8^{2}$ .

$f (x) = \frac{(x - 9) (x + 7)}{x^{2} - 8^{2}}$

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 $b = 8$ .

$f (x) = \frac{(x - 9) (x + 7)}{(x + 8) (x - 8)}$