"Minus times minus gives plus - the root was there all the time and is not extraneous."
Regarding the extraneous root, I was referring not to the derivation of Golden Ratio, but to the other problem given on my page:
Sqrt(1 + Sqrt(1 + Sqrt(1 + ...)))
In this case, there is only plusses. The square root is therefore positive. Here's an vastly simplified example of what I'm talking about:
Say we way have y=sqrt(5). In this case, the notation means y is equal to the positive sqrt of 5 only. We know that we can square both sides, so we do this:
y^2 = 5, BUT the moment we did this we introduced an extraneous root, -sqrt(5)!
Now, in general, this is not extraneous, but in this particular case, we know that the result is ONLY the positive square root (by definition).
I very clearly understand your point on the triangles. Sometimes it is appropriate to consider negative lengths, and other times it is not. It depends on the particulars of the situation. Generally speaking, one doesn't consider negative lengths for the sides of a triangle. Often, I think of distances being negative, but lengths being only positive, but I'm not sure that's correct.
Unfortunatley, while I use algebra, geometry, trig, analytic geometry almost every day at work, I don't often put a lot of thought into why I reject some solutions. The bottom line is I look at it and I just know whether it makes sense or not. I don't have a firm and formal grasp on why I reject. The only criteria is - does it make sense in the context of the particular problem?
k
