First there is a typo in your post, the phrase:

"consists of all numbers x for which -(sqrt{a}) < a (sqrt{a})"

is probably mistyped as it is always true, there should be an x in there somewhere.

If you sketch the graph of x^2-a (with a>0) you will see that this is >0 everywhere

except between the roots of x^2-a, and when this is >0 we have x^2>a.

So x^2>0 when x<-sqrt(a), or when x>sqrt(a).

RonL