Algebra Examples

$\frac{2 q^{2} - 4 q}{4 q^{2} - 36} - \frac{6}{4 q^{2} - 36}$

Step 1

Simplify terms.

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Step 1.1

Combine the numerators over the common denominator.

$\frac{2 q^{2} - 4 q - 6}{4 q^{2} - 36}$

Step 1.2

Cancel the common factor of $2 q^{2} - 4 q - 6$ and $4 q^{2} - 36$ .

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Step 1.2.1

Factor $2$ out of $2 q^{2}$ .

$\frac{2 (q^{2}) - 4 q - 6}{4 q^{2} - 36}$

Step 1.2.2

Factor $2$ out of $- 4 q$ .

$\frac{2 (q^{2}) + 2 (- 2 q) - 6}{4 q^{2} - 36}$

Step 1.2.3

Factor $2$ out of $2 (q^{2}) + 2 (- 2 q)$ .

$\frac{2 (q^{2} - 2 q) - 6}{4 q^{2} - 36}$

Step 1.2.4

Factor $2$ out of $- 6$ .

$\frac{2 (q^{2} - 2 q) + 2 \cdot - 3}{4 q^{2} - 36}$

Step 1.2.5

Factor $2$ out of $2 (q^{2} - 2 q) + 2 \cdot - 3$ .

$\frac{2 (q^{2} - 2 q - 3)}{4 q^{2} - 36}$

Step 1.2.6

Cancel the common factors.

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Step 1.2.6.1

Factor $2$ out of $4 q^{2}$ .

$\frac{2 (q^{2} - 2 q - 3)}{2 (2 q^{2}) - 36}$

Step 1.2.6.2

Factor $2$ out of $- 36$ .

$\frac{2 (q^{2} - 2 q - 3)}{2 (2 q^{2}) + 2 (- 18)}$

Step 1.2.6.3

Factor $2$ out of $2 (2 q^{2}) + 2 (- 18)$ .

$\frac{2 (q^{2} - 2 q - 3)}{2 (2 q^{2} - 18)}$

Step 1.2.6.4

Cancel the common factor.

$\frac{2 (q^{2} - 2 q - 3)}{2 (2 q^{2} - 18)}$

Step 1.2.6.5

Rewrite the expression.

$\frac{q^{2} - 2 q - 3}{2 q^{2} - 18}$

Step 2

Factor $q^{2} - 2 q - 3$ 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 $- 3$ and whose sum is $- 2$ .

$- 3, 1$

Step 2.2

Write the factored form using these integers.

$\frac{(q - 3) (q + 1)}{2 q^{2} - 18}$

Step 3

Simplify the denominator.

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Step 3.1

Factor $2$ out of $2 q^{2} - 18$ .

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Step 3.1.1

Factor $2$ out of $2 q^{2}$ .

$\frac{(q - 3) (q + 1)}{2 (q^{2}) - 18}$

Step 3.1.2

Factor $2$ out of $- 18$ .

$\frac{(q - 3) (q + 1)}{2 q^{2} + 2 \cdot - 9}$

Step 3.1.3

Factor $2$ out of $2 q^{2} + 2 \cdot - 9$ .

$\frac{(q - 3) (q + 1)}{2 (q^{2} - 9)}$

Step 3.2

Rewrite $9$ as $3^{2}$ .

$\frac{(q - 3) (q + 1)}{2 (q^{2} - 3^{2})}$

Step 3.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = q$ and $b = 3$ .

$\frac{(q - 3) (q + 1)}{2 (q + 3) (q - 3)}$

Step 4

Cancel the common factor of $q - 3$ .

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Step 4.1

Cancel the common factor.

$\frac{(q - 3) (q + 1)}{2 (q + 3) (q - 3)}$

Step 4.2

Rewrite the expression.

$\frac{q + 1}{2 (q + 3)}$