Calculator Inputs
Use the form below. Results appear above this form after submission.
Formula Used
The discriminant of a quadratic equation ax² + bx + c = 0 is calculated with:
If D > 0: the equation has two real and distinct roots.
If D = 0: the equation has one repeated real root.
If D < 0: the equation has two complex conjugate roots.
Helpful companion formulas: axis of symmetry = -b / (2a), and the repeated or real roots come from x = (-b ± √D) / (2a).
How to Use This Calculator
- Enter values for a, b, and c from your quadratic equation.
- Set a graph range using minimum x and maximum x.
- Choose how many plot points you want for the curve.
- Select the decimal precision for displayed results.
- Press Calculate Discriminant.
- Review the result card above the form.
- Inspect the graph, roots, axis, and vertex.
- Export the current result as CSV or PDF when needed.
Example Data Table
| # | a | b | c | Discriminant | Nature of roots | Roots |
|---|---|---|---|---|---|---|
| 1 | 1 | -5 | 6 | 1 | Two real distinct | 2, 3 |
| 2 | 1 | 4 | 4 | 0 | One repeated | -2 |
| 3 | 2 | 1 | 5 | -39 | Complex pair | -0.25 ± 1.5612i |
| 4 | 3 | -12 | 12 | 0 | One repeated | 2 |
| 5 | 1 | 0 | -9 | 36 | Two real distinct | -3, 3 |
Frequently Asked Questions
1) What is the discriminant in a quadratic equation?
The discriminant is the value b² - 4ac. It tells you how many roots a quadratic has and whether those roots are real, repeated, or complex.
2) What does a positive discriminant mean?
A positive discriminant means the quadratic has two different real roots. On the graph, the parabola crosses the x-axis at two separate points.
3) What happens when the discriminant equals zero?
When the discriminant is zero, the quadratic has one repeated real root. The parabola touches the x-axis exactly once at its vertex.
4) What does a negative discriminant mean?
A negative discriminant means there are no real x-intercepts. The equation has two complex conjugate roots, and the graph stays above or below the x-axis.
5) Why must coefficient a be nonzero?
If a equals zero, the equation is no longer quadratic. It becomes linear, so the discriminant rule for quadratics does not apply.
6) Does the calculator only classify roots?
No. It also gives the discriminant value, axis of symmetry, vertex, parabola direction, the exact root style, and a graph of the quadratic.
7) Why is the graph useful here?
The graph helps you connect the algebra to the shape of the parabola. You can quickly see crossings, turning points, and symmetry.
8) Can I use this calculator for homework checking?
Yes. It is useful for checking class exercises, verifying manual calculations, and comparing how different coefficients change the discriminant and root behavior.