How can you evaluate $\frac{x}{x^{2} - 4} - \frac{2}{x^{2} - 4}$ $(x)/(x^2-4) - (2)/(x^2-4)$ ?

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Answer 1 · 2015-07-22T15:53:53

$= \frac{1}{x + 2}$ $=color(blue)(1/(x+2)$

with exclusion $x \neq 2$ $x != 2$

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

Here the denominators are the same, so combining the numerators we get:

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

As per property:
$a^{2} - b^{2} = (a + b) (a - b)$ $color(blue)(a^2-b^2) = (a+b)(a-b)$

Similarly :
$(x^{2} - 4) = (x + 2) (x - 2)$ $color(blue)((x^2-4)) =(x+2)(x-2)$

Expressing the denominator in the this way, the expression becomes:

$cancel(x- 2)/color(blue)((x+2)cancel((x-2))$

$=color(blue)(1/(x+2)$

with exclusion $x != 2$

How can you evaluate (x)/(x^2-4) - (2)/(x^2-4)xx2−4−2x2−4(x)/(x^2-4) - (2)/(x^2-4)?