I recently discover the cool proof of the Cauchy Schwarz inequality : $$ |\langle x, y\rangle| \le \Vert x \Vert \Vert y \Vert$$ for all $ x, y \in \Bbb R^n$, using the discriminant : $$ P(t) = \Vert x+ty \Vert^2 = \Vert x \Vert^2 + 2t\langle x,y \rangle + t^2\Vert y \Vert^2 $$ This polynomial is positive by definition of the inner product, and $ \Vert y \Vert^2 $ is positive, thus the discriminant is less or equal to zero : $$ 4 \langle x, y\rangle^2 - 4\Vert x \Vert^2\Vert y \Vert^2 \le 0$$ giving the wanted inequality.
Do you know any other proofs that use this discriminant cool trick ?