simplify fraction with radical in numerator, first steps?

Question

$(\sqrt x−2)/(x−4)$

Looking to simplify a fraction with radical in the numerator.

Answer 1

$x-4=(\sqrt{x})^2-(2)^2=(\sqrt{x}+2)(\sqrt{x}-2)$.

Can you take it from here.

Answer 2

$$\frac {\sqrt x−2}{(x−4)}$$

its already simplified

if you do

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

then you will get

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

which is not simplified form

Example; for x=2

$$\frac {\sqrt x−2}{(x−4)}$$

$$=\frac {\sqrt 2−2}{(2−4)}$$

$$=\frac {\sqrt 2−2}{(-2)}$$

which is easy to solve

for x=2

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

$$\frac {1}{(\sqrt{2}+2)}$$

which is not easy to solve

Answer 3

It may help to work with this in terms of $t = \sqrt x$. You'll more readily see how we can factor a difference of squares here:

$$\begin{align} \frac{\sqrt x - 2}{x - 4} \quad & \overset{t\; = \sqrt x}{=} \quad\frac{t - 2}{t^2 - 4} \\ \\ & \quad = \dfrac{t - 2}{(t-2)(t+2)} \\ \\ & \quad= \dfrac 1{t+2} \\ \\ & \overset{t\;= \sqrt x}= \dfrac 1{\sqrt x + 2}\end{align}$$