The discriminant and the roots

The discriminant

To solve ax^2 + bx + c = 0, we calculate the discriminant:

delta = b^2 - 4ac

This number indicates how many real solutions there are.

The three cases

Sign of delta	Number of solutions	Formulas
delta > 0	2 distinct solutions	x₁ = (-b - √(delta)) / (2a); x₂ = (-b + √(delta)) / (2a)
delta = 0	1 double root	x₀ = -b / (2*a)
delta < 0	no real solutions	-

Complete example

Let’s solve x² - 5x + 6 = 0. a = 1, b = -5, c = 6. delta = (-5)² - 4 × 1 × 6 = 25 - 24 = 1. delta > 0, so there are two solutions: x₁ = (5 - √1) / 2 = 4/2 = 2 x₂ = (5 + √1) / 2 = 6/2 = 3

Another example: delta < 0

For x² + x + 1 = 0: delta = 1 - 4 = -3 < 0. No real solutions exist: the curve never intersects the x-axis.

Common pitfall

Never forget the minus sign in front of 4ac: delta = b² - 4ac, not b² + 4ac. Similarly, when b is negative, -b becomes positive: for b = -5, -b = 5.

The discriminant is the key tool in this chapter: make sure you’ve mastered it before moving on to factorisation.