The function in the proof of Cauchy-Schwarz has complex zeros, so set discriminant less than zero

Quadratic Equation nature of the solution analysis depends on the algebraic sign of the discriminant and trichotomy. A quadratic expression with complex solutions has a negative discriminant which is the basis for the proof of the infamous Cauchy-Schwarz inequality.
Math teacher Homer Colunga(Rate My Professor favorite-Ryanmccx1)

comprehensively and lucidly covered complex zeroes of polynomials.