Year 10: The quadratic equation discriminant

This cheat-sheet explains the discriminant of a quadratic equation. The discriminant helps you determine the nature of the solutions (roots) of the equation.

What is the Discriminant?

A quadratic equation is in the standard form: ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0.

The discriminant (often denoted as Δ or D) is calculated using the following formula:

Δ = b² - 4ac

Interpreting the Discriminant

• Δ > 0: The quadratic equation has two distinct real roots. This means the parabola represented by the equation intersects the x-axis at two points.
• Δ = 0: The quadratic equation has one real root (or two equal real roots). This means the parabola touches the x-axis at exactly one point (the vertex of the parabola lies on the x-axis).
• Δ < 0: The quadratic equation has no real roots. This means the parabola does not intersect the x-axis. The roots are complex numbers.

Example

Consider the equation: x² + 5x + 6 = 0

Δ = 5² - 4(1)(6) = 25 - 24 = 1

Since Δ > 0, the equation has two distinct real roots.