Algebra Examples

$\frac{x^{2} - 5 x - 14}{x^{2} - 49}$

Step 1

Factor $x^{2} - 5 x - 14$ 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 $- 14$ and whose sum is $- 5$ .

$- 7, 2$

Step 1.2

Write the factored form using these integers.

$\frac{(x - 7) (x + 2)}{x^{2} - 49}$

Step 2

Rewrite $49$ as $7^{2}$ .

$\frac{(x - 7) (x + 2)}{x^{2} - 7^{2}}$

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 $b = 7$ .

$\frac{(x - 7) (x + 2)}{(x + 7) (x - 7)}$

Step 4

Reduce the expression $\frac{(x - 7) (x + 2)}{(x + 7) (x - 7)}$ by cancelling the common factors.

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Step 4.1

Cancel the common factor.

$\frac{(x - 7) (x + 2)}{(x + 7) (x - 7)}$

Step 4.2

Rewrite the expression.

$\frac{x + 2}{x + 7}$