Stuck on the first chapter of Apostol:

[Gwyn]

I've been going over the book and everything was going well until I got to this question:

$$\frac{1} {2 + \sqrt{4 - x^2}} = \frac{2 - \sqrt{4 - x^2}} {x^2}$$

Tried with partial fractions but that does not work. I also tried to split the fraction but I don't know how to get the x^2 at the bottom. In short I'm completely lost and that looks like black magic to me. I'd appreciate it if anyone has a link to explain what's going on here.

Edit: Messed up the latex. If you click on it the correct code shows up. Not sure how I can refresh the originally posted one.

 

[courtrigrad]

He just rationalized the denominator.

$$\frac{1} {2 + \sqrt{4 - x^2}} \ \frac{2-\sqrt{4 - x^2}}{2-\sqrt{4 - x^2}} = \frac{2 - \sqrt{4 - x^2}} {x^2}$$

Note the denominator is a difference of squares: $$a^{2}-b^{2}$$

 

[Gwyn]

Thanks a lot, I was stuck on that for a long time.