Algebra Examples

$\frac{x^{2} + x - 6}{x^{2} - 4}$

Step 1

Factor $x^{2} + x - 6$ 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 $- 6$ and whose sum is $1$ .

$- 2, 3$

Step 1.2

Write the factored form using these integers.

$\frac{(x - 2) (x + 3)}{x^{2} - 4}$

Step 2

Rewrite $4$ as $2^{2}$ .

$\frac{(x - 2) (x + 3)}{x^{2} - 2^{2}}$

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 $b = 2$ .

$\frac{(x - 2) (x + 3)}{(x + 2) (x - 2)}$

Step 4

Reduce the expression $\frac{(x - 2) (x + 3)}{(x + 2) (x - 2)}$ by cancelling the common factors.

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Step 4.1

Cancel the common factor.

$\frac{(x - 2) (x + 3)}{(x + 2) (x - 2)}$

Step 4.2

Rewrite the expression.

$\frac{x + 3}{x + 2}$