>> But why is that, exactly? Well, looking in my sister's A-level maths 
>> book [unlike me, my sister was *taught* mathematics] I discover a long 
>> sequence of algebraic manipulations that slowly transforms the one 
>> into the other.
> 
> It's not that hard or long, just involves dividing by a, completing the 
> square for the first two terms, rearranging, taking the square root, and 
> rearranging again for x.

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.)

I also note in passing that I have no clue what "completing the square" 
actually means. But then, I have no formal mathematical education at 
all, so it's a miracle I even comprehend algebra in the first place!

> But I imagine the cubic is a lot harder.

http://planetmath.org/?op=getobj&from=objects&id=1407

Does it make your eyes bleed?

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