Algebra Examples

$\frac{3 x^{2} - 12}{x^{2} - x - 6}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $3$ out of $3 x^{2} - 12$ .

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Step 1.1.1

Factor $3$ out of $3 x^{2}$ .

$\frac{3 (x^{2}) - 12}{x^{2} - x - 6}$

Step 1.1.2

Factor $3$ out of $- 12$ .

$\frac{3 x^{2} + 3 \cdot - 4}{x^{2} - x - 6}$

Step 1.1.3

Factor $3$ out of $3 x^{2} + 3 \cdot - 4$ .

$\frac{3 (x^{2} - 4)}{x^{2} - x - 6}$

Step 1.2

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

$\frac{3 (x^{2} - 2^{2})}{x^{2} - x - 6}$

Step 1.3

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 $b = 2$ .

$\frac{3 (x + 2) (x - 2)}{x^{2} - x - 6}$

Step 2

Factor $x^{2} - x - 6$ using the AC method.

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Step 2.1

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 $- 1$ .

$- 3, 2$

Step 2.2

Write the factored form using these integers.

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

Step 3

Cancel the common factor of $x + 2$ .

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Step 3.1

Cancel the common factor.

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

Step 3.2

Rewrite the expression.

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