Simplify x/(x-2)+-8/(x^2-4)

$\frac{x}{x - 2} + \frac{- 8}{x^{2} - 4}$

Step 1

Simplify each term.

Simplify the denominator.

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

$\frac{x}{x - 2} + \frac{- 8}{x^{2} - 2^{2}}$

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 $b = 2$ .

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

Move the negative in front of the fraction.

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

Step 2

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

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

Step 3

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

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

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

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

Apply the distributive property.

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

Multiply $x$ by $x$ .

$\frac{x^{2} + x \cdot 2 - 8}{(x + 2) (x - 2)}$

Move $2$ to the left of $x$ .

$\frac{x^{2} + 2 \cdot x - 8}{(x + 2) (x - 2)}$

Factor $x^{2} + 2 x - 8$ 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 $- 8$ and whose sum is $2$ .

$- 2, 4$

Write the factored form using these integers.

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

Step 6

Cancel the common factor of $x - 2$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{x + 4}{x + 2}$

Simplify x/(x-2)+-8/(x^2-4)

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