A quadratic equation in one unknown has the form \(ax^2 + bx + c = 0\), where \(a \neq 0\). You can solve these by factorisation, the quadratic formula, or by reading information from the graph of \(y = ax^2 + bx + c\).
The discriminant \(\Delta\) (delta) is defined below. It tells you how many real roots the equation has without fully solving it.
Definition
\[ \Delta = b^2 - 4ac \]
| Discriminant | Nature of roots | Graph of \(y = ax^2 + bx + c\) |
|---|---|---|
| \(\Delta > 0\) | Two distinct real roots | Parabola cuts the x-axis twice |
| \(\Delta = 0\) | One repeated real root (equal roots) | Vertex touches the x-axis |
| \(\Delta < 0\) | No real roots | Parabola lies entirely above or below the x-axis |
When \(\Delta \geq 0\), the quadratic formula gives the roots: \[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \]
Use the explorer below to change \(a\), \(b\), and \(c\). Watch how \(\Delta\) and the parabola change, and read off the nature of the roots.