The discriminant is the expression for a quadratic equation , where . Its sign tells you whether the quadratic has two distinct real roots, two equal real roots, or no real roots.
The discriminant is the expression for a quadratic equation , where . Its sign tells you whether the quadratic has two distinct real roots, two equal real roots, or no real roots.
The quadratic formula is
The expression under the square root is the discriminant. Therefore, the possible roots depend on whether is a positive real number, zero, or not real.
| Value of the discriminant | Nature of the roots | Graphical meaning |
|---|---|---|
| Two distinct real roots | The parabola crosses the -axis twice | |
| Two equal real roots, giving one repeated root | The parabola touches the -axis once | |
| No real roots | The parabola does not meet the -axis |
For example, consider . Here, , , and , so
Since , the equation has two distinct real roots. In fact, factorization gives , so the roots are and .
Compare two further cases. For , , so there is one repeated root, ; graphically, the parabola is tangent to the -axis. For , , so there are no real roots and no -intercepts.
In parameter questions, translate the required root type into , , or , then solve the resulting inequality or equation. Also check , because otherwise the expression is not quadratic. This links the algebraic test directly to the graph's -intercepts.
A common misconception is that the discriminant gives the values of the roots. It does not: it gives their number and nature. To calculate the roots themselves, use factorization, completing the square, or the quadratic formula.
This is IB Mathematics AA SL 2.7 and may be tested on either Paper 1 or Paper 2. For a command such as “determine the nature of the roots,” state , substitute correctly, compare the result with zero, and give a written conclusion. Do not stop after calculating the discriminant.