Use the discriminant for a quadratic equation in the form , where . Its sign tells you immediately whether the quadratic has two distinct real roots, one repeated real root, or no real roots.
Use the discriminant for a quadratic equation in the form , where . Its sign tells you immediately whether the quadratic has two distinct real roots, one repeated real root, or no real roots.
This is part of IB Mathematics: Analysis and Approaches SL 2.7: Solutions of quadratic equations and inequalities, discriminant and nature of roots. It is core content for both SL and HL students and may appear on Paper 1 or Paper 2.
The quadratic formula is
The expression inside the square root is the discriminant. Its sign determines whether the square root produces two real values, one real value, or no real values.
| Value of the discriminant | Nature of the roots | Graphical meaning |
|---|---|---|
| Two distinct real roots | The parabola crosses the -axis twice. | |
| Two equal real roots, also called one repeated root | The parabola touches the -axis once at its vertex. | |
| No real roots | The parabola does not meet the -axis. |
For example, consider . Here , , and , so
Since , the equation has two distinct real roots.
Compare two further cases. For , the discriminant is , so there is one repeated real root, . For , the discriminant is , so there are no real roots. These examples show that only the sign of , not its size, controls the classification. The magnitude does not tell you how far apart the roots are.
A common misconception is that the discriminant gives the roots themselves. It does not: it gives the nature of the roots. To find their actual values, factorize, complete the square, or use the quadratic formula.
In an IB exam, write , substitute correctly, and then state a conclusion linked to its sign. Do not give only the calculated discriminant when the command term asks you to determine or state the nature of the roots.