Simplify 5/(2x-12)-20/(x^2-4x-12)

$\frac{5}{2 x - 12} - \frac{20}{x^{2} - 4 x - 12}$

Step 1

Simplify each term.

Factor $2$ out of $2 x - 12$ .

Factor $2$ out of $2 x$ .

$\frac{5}{2 (x) - 12} - \frac{20}{x^{2} - 4 x - 12}$

Factor $2$ out of $- 12$ .

$\frac{5}{2 x + 2 \cdot - 6} - \frac{20}{x^{2} - 4 x - 12}$

Factor $2$ out of $2 x + 2 \cdot - 6$ .

$\frac{5}{2 (x - 6)} - \frac{20}{x^{2} - 4 x - 12}$

Factor $x^{2} - 4 x - 12$ using the AC method.

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 $- 12$ and whose sum is $- 4$ .

$- 6, 2$

Write the factored form using these integers.

$\frac{5}{2 (x - 6)} - \frac{20}{(x - 6) (x + 2)}$

Step 2

To write $\frac{5}{2 (x - 6)}$ as a fraction with a common denominator, multiply by $\frac{x + 2}{x + 2}$ .

$\frac{5}{2 (x - 6)} \cdot \frac{x + 2}{x + 2} - \frac{20}{(x - 6) (x + 2)}$

Step 3

To write $- \frac{20}{(x - 6) (x + 2)}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{5}{2 (x - 6)} \cdot \frac{x + 2}{x + 2} - \frac{20}{(x - 6) (x + 2)} \cdot \frac{2}{2}$

Step 4

Write each expression with a common denominator of $2 (x - 6) (x + 2)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{5}{2 (x - 6)}$ by $\frac{x + 2}{x + 2}$ .

$\frac{5 (x + 2)}{2 (x - 6) (x + 2)} - \frac{20}{(x - 6) (x + 2)} \cdot \frac{2}{2}$

Multiply $\frac{20}{(x - 6) (x + 2)}$ by $\frac{2}{2}$ .

$\frac{5 (x + 2)}{2 (x - 6) (x + 2)} - \frac{20 \cdot 2}{(x - 6) (x + 2) \cdot 2}$

Reorder the factors of $2 (x - 6) (x + 2)$ .

$\frac{5 (x + 2)}{2 (x + 2) (x - 6)} - \frac{20 \cdot 2}{(x - 6) (x + 2) \cdot 2}$

Reorder the factors of $(x - 6) (x + 2) \cdot 2$ .

$\frac{5 (x + 2)}{2 (x + 2) (x - 6)} - \frac{20 \cdot 2}{2 (x + 2) (x - 6)}$

Step 5

Combine the numerators over the common denominator.

$\frac{5 (x + 2) - 20 \cdot 2}{2 (x + 2) (x - 6)}$

Step 6

Simplify the numerator.

Factor $5$ out of $5 (x + 2) - 20 \cdot 2$ .

Factor $5$ out of $- 20 \cdot 2$ .

$\frac{5 (x + 2) + 5 (- 4 \cdot 2)}{2 (x + 2) (x - 6)}$

Factor $5$ out of $5 (x + 2) + 5 (- 4 \cdot 2)$ .

$\frac{5 (x + 2 - 4 \cdot 2)}{2 (x + 2) (x - 6)}$

Multiply $- 4$ by $2$ .

$\frac{5 (x + 2 - 8)}{2 (x + 2) (x - 6)}$

Subtract $8$ from $2$ .

$\frac{5 (x - 6)}{2 (x + 2) (x - 6)}$

Step 7

Cancel the common factor of $x - 6$ .

Cancel the common factor.

$\frac{5 (x - 6)}{2 (x + 2) (x - 6)}$

Rewrite the expression.

$\frac{5}{2 (x + 2)}$

Simplify 5/(2x-12)-20/(x^2-4x-12)

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