I'm working through Spivak's Calculus over the summer, and I'm currently on problem 19 of Chapter 1, which involves proving the Schwarz inequality. The first two parts of the proof are fairly straightforward, but I don't understand the last part. Spivak gives the inequality:
and suggests using the quadratic formula (I'm not sure I understand this part, since the equation is greater than zero, how can there be any solutions?) to arrive at the Schwarz inequality. In the answer key, the following is given.
I'm not really sure how this follows since I can't get it just by trying to find the roots of the equations vis a vis the quadratic formula.