Intuition behind Algebra When proving, or just straight up solving equations, we often manipulate variables until we get the results we want. For example, the square root of the discriminant is simply an algebraic manipulation to get the result of the distance (or difference) between the roots. However, is there intuition behind this manipulation? I get why Algebra works in the first place but when seeing it in action, manipulations as complex as that of involved in getting the discriminant, gets me lose sense of what's actually happening within the numbers.

In the particular example you mentioned, you can understand what's going on by looking at the graph of the parabola with equation $y = ax^2 + bx + c$. It's not hard to see that the vertex of the parabola is at the point where 
$$
x = - \frac{b}{2a} \quad ; \quad y = -\frac{b^2-4ac}{4a}
$$
There's a better explanation here. So, you can understand the algebra by looking at the geometry. And (for me, anyway), developing intuition about geometry is not too hard because I can draw pictures. 
On the other hand, there are algebraic processes for which it's difficult to draw pictures. Or, at least, I don't know how to draw them (I'd love to learn). In those cases, the algebra becomes (for me) a mysterious process of shuffling symbols around, which is devoid of any intuition.
Different people have different types of "intuition", and develop it in different ways. My way (drawing pictures) might not be the best way for you.
