Simplify 4/(x^2-4)-1/(x-2)

$\frac{4}{x^{2} - 4} - \frac{1}{x - 2}$

Step 1

Simplify the denominator.

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

$\frac{4}{x^{2} - 2^{2}} - \frac{1}{x - 2}$

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 $b = 2$ .

$\frac{4}{(x + 2) (x - 2)} - \frac{1}{x - 2}$

Step 2

To write $- \frac{1}{x - 2}$ as a fraction with a common denominator, multiply by $\frac{x + 2}{x + 2}$ .

$\frac{4}{(x + 2) (x - 2)} - \frac{1}{x - 2} \cdot \frac{x + 2}{x + 2}$

Step 3

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

Multiply $\frac{1}{x - 2}$ by $\frac{x + 2}{x + 2}$ .

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

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

Apply the distributive property.

$\frac{4 - x - 1 \cdot 2}{(x + 2) (x - 2)}$

Multiply $- 1$ by $2$ .

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

Subtract $2$ from $4$ .

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

Step 6

Reduce the expression by cancelling the common factors.

Cancel the common factor of $- x + 2$ and $x - 2$ .

Factor $- 1$ out of $- x$ .

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

Rewrite $2$ as $- 1 (- 2)$ .

$\frac{- (x) - 1 (- 2)}{(x + 2) (x - 2)}$

Factor $- 1$ out of $- (x) - 1 (- 2)$ .

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

Rewrite $- (x - 2)$ as $- 1 (x - 2)$ .

$\frac{- 1 (x - 2)}{(x + 2) (x - 2)}$

Cancel the common factor.

$\frac{- 1 (x - 2)}{(x + 2) (x - 2)}$

Rewrite the expression.

$\frac{- 1}{x + 2}$

Move the negative in front of the fraction.

$- \frac{1}{x + 2}$