SECTION 3.5 Quadratic Discriminant 161
3.5 The Discriminant of a Quadratic
The discriminant of a quadratic polynomial, while finding itself in
(mostly trivial) discussions in a typical high-school Algebra II course,
nonetheless is a highly underused and too narrowly understood concept.
This and the next two sections will attempt to provide meaningful ap-
plications of the discriminant, as well as put it in its proper algebraic
perspective. Before proceeding, let me remind the reader that a possi-
bly surprising application of the discriminant has already occurred in
the proof of the Cauchy-Schwarz inequality (page 150).
Given the quadratic polynomialf(x) =ax^2 +bx+c, a,b,c∈R, the
discriminantis defined by the familiar recipe:
D = b^2 − 4 ac.This expression is typically introduced as a by-product of the quadratic
formula expressing the two rootsα, βof the equationf(x) = 0 as
α, β =
−b±√
b^2 − 4 ac
2 a=
−b±√
D
2 a.
From the above, the following simple trichotomy emerges.
D > 0 ⇐⇒f(x) = 0 has two distinct real roots;D < 0 ⇐⇒f(x) = 0 has two imaginary conjugate roots;D= 0⇐⇒f(x) = 0 has a double real root.Note that if f(x) = ax^2 +bx+c with a >0, then the condition
D≤0 implies the unconditional inequalityf(x)≥0.