The discriminant and the roots of the trinomial

The discriminant

For f(x) = ax^2 + bx + c (where a is not equal to 0), the discriminant is calculated as follows:

Delta = b^2 - 4ac

The sign of Δ indicates the number of roots (the solutions to the equation f(x) = 0).

Sign of Δ	Number of roots	Roots
Δ > 0	2 distinct roots	x₁ = (-b - √Δ) / (2a), x₂ = (-b + √Δ) / (2a)
Delta = 0	1 double root	x0 = -b / (2*a)
Delta < 0	No real roots	-

Example

Let f(x) = x² - 5x + 6.

a = 1, b = -5, c = 6

Δ = (-5)² - 4 × 1 × 6 = 25 - 24 = 1

Δ > 0, so there are two roots:

x₁ = (5 – 1) / 2 = 2

x₂ = (5 + 1) / 2 = 3

Common pitfall

Don’t forget the minus sign in front of 4ac: Delta = b^2 - 4ac, not b^2 + 4ac. A mistake with the sign completely changes the result.

Another tip: if c = 0, you can factor directly by x without calculating Delta: f(x) = x*(a*x + b).

Key points

The discriminant is the key to any sign analysis: it must be calculated first, as the method that follows depends on its sign.