Year 10: The quadratic equation discriminant

This cheat-sheet explains the discriminant of a quadratic equation. It's a key tool for understanding the nature of the solutions (roots) of a quadratic equation.

What is the Discriminant?

A quadratic equation is generally written in the form: f(x) = ax² + bx + c, where a, b, and c are constants, and a ≠ 0.

The discriminant, denoted by the symbol Δ (delta), is calculated as follows:

Δ = b² - 4ac

Interpreting the Discriminant

The value of Δ tells you how many real solutions the quadratic equation has:

• Δ > 0: Two distinct real solutions. This means the parabola represented by the quadratic equation crosses the x-axis twice.
• Δ = 0: One real solution (a repeated root). This means the parabola touches the x-axis at exactly one point (the vertex of the parabola is on the x-axis).
• Δ < 0: No real solutions. This means the parabola does not intersect the x-axis. The solutions are complex numbers.

Example

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

Here, a = 1, b = -5, and c = 6.

Δ = (-5)² - 4(1)(6) = 25 - 24 = 1

Since Δ > 0, there are two distinct real solutions.