> Yeah. As I said, a sequence of algebraic operations which "just happen" to 
> lead to the answer we seek. (As in, there's no *apparent* reason why this 
> particular sequence of operations does it, as opposed to some other.)

Maybe if you've never done any equation solving before, but the steps are 
pretty obvious steps to take to rearrange an equation like that (ie to get 
all the x's in one place).

> I also note in passing that I have no clue what "completing the square" 
> actually means.

It's the key step to rearranging the quadratic equation in terms of x, and 
useful in lots of other areas too.  It's the process of rewriting terms like 
x^2+cx in the form (x+a)^2 + b.  It's not hard, and after performing a 
square root allows you to get the x by itself.

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