Originally posted by DoctorScribbles
I hope that:
1. That was not the method used, or
2. If not (1), then she realizes that that method fails in general due to the instances of polynomials in which a=1 is only one solution among many, and
3. If (2) and not (1), then she acknowledges that she just got lucky this time.
Math is a war, not a battle!
In such a trivial instance, it doesn't really matter. In general, general methods are useful, and even essential, to know, but it's more important to know the details of what goes into the method (the facts about what a tangent is, how it relates to the derivative, etc.). Thus in the War, we choose from our mathematical knowledge what we need to knkow to construct a method. In an easy battle, we choose as few facts as possible and construct a simple, ad hoc, possibly elegant method. In a hard battle, we construct a powerful, general, complicated method. Part of the reason for this is aesthetics, part is the fact that sometimes, if we want to be efficient, well-developed intution helps.