The sign rule for a trinomial

General principle

The sign of f(x) = ax^2 + bx + c depends on two things: the sign of a and the number of roots (given by Δ).

Case where Δ < 0

f(x) has no real roots: the trinomial retains a constant sign, that of a, for all real x.

Example: f(x) = x² + x + 1, Δ = 1 – 4 = –3 < 0, a = 1 > 0, so f(x) > 0 for all x.

Case where Delta = 0

f(x) has a double root x₀. The trinomial retains the sign of a everywhere, except at x₀ where it is zero.

Case where Δ > 0

f(x) has two distinct roots x₁ < x₂. Rule to remember:

“f(x) has the sign of a outside the roots, and the sign of –a (the opposite of a) between the roots.”

Interval	Sign of f(x)
x < x₁	sign of a
x₁ < x < x₂	sign of -a
x > x₂	sign of a

Example

f(x) = x² - 5x + 6, roots x₁ = 2, x₂ = 3, a = 1 > 0.

f(x) > 0 on ]∞; 2[ ∪ ]3; +∞[

f(x) < 0 on ]2; 3[

f(x) = 0 at x = 2 and x = 3

Common pitfall

Many pupils forget to reverse the sign between the roots. Remember this mnemonic: “sign of a, except between the roots”.