Discriminant Calculator - Quadratic Root Analysis
Calculate the discriminant Δ = b² − 4ac of any quadratic equation and instantly determine whether its roots are real, repeated, or complex.
Discriminant Calculator - Quadratic Root Analysis
Calculate the discriminant Δ = b² − 4ac of any quadratic equation and instantly determine whether its roots are real, repeated, or complex.
Enter coefficients a, b, and c from ax² + bx + c = 0. The coefficient a cannot be zero.
Load a quick example:
About the Discriminant Calculator
The discriminant is a single number that encodes everything you need to know about the roots of a quadratic equation before solving it. Derived from the quadratic formula, the discriminant Δ = b² − 4ac sits under the square root sign in x = (−b ± √Δ) / (2a). Its sign alone determines whether the equation has two distinct real roots (Δ > 0), one repeated real root (Δ = 0), or two complex conjugate roots (Δ < 0).
When Δ is positive, its square root is a real positive number, and the ± in the quadratic formula produces two different real values. The larger root is (−b + √Δ)/(2a) and the smaller is (−b − √Δ)/(2a). A larger discriminant means the two roots are farther apart; the roots are closest when Δ is small and positive. On the graph of y = ax² + bx + c, a positive discriminant means the parabola crosses the x-axis at two distinct points.
When Δ equals zero, √Δ = 0, and both the + and − branches give the same answer: x = −b/(2a). This is the vertex of the parabola, and the curve is tangent to the x-axis at exactly this one point. Perfect square trinomials like (x − 3)² = x² − 6x + 9 always have discriminant zero: Δ = 36 − 36 = 0.
When Δ is negative, there is no real square root of Δ, and the solutions involve the imaginary unit i = √(−1). The two roots are complex conjugates of the form (−b)/(2a) ± i√(|Δ|)/(2a). While these roots do not correspond to x-intercepts on the real number line, they are genuine solutions in the complex number system and appear frequently in signal processing, control theory, and physics.
The discriminant has important connections to other parts of mathematics. In the quadratic formula, it determines the two solutions directly. In coordinate geometry, it controls the position of the parabola relative to the x-axis. In the theory of equations, it generalises to higher-degree polynomials as a measure of how many roots coincide. Vieta's formulas connect the discriminant to the sum and product of roots: for ax² + bx + c = 0, the sum of roots is −b/a and the product is c/a, and Δ = (sum of roots)² − 4(product of roots) × a²/a² in normalised form.
Enter any valid a, b, c into the discriminant calculator to instantly see Δ, the nature of the roots, and the actual root values. The calculator handles all three cases — positive, zero, and negative discriminant — and presents complex roots in the standard a + bi form.
Discriminant Examples
Three standard cases covering every possible discriminant outcome.
| Equation | Discriminant | Root Nature |
|---|---|---|
| x² − 5x + 6 = 0 (a=1, b=−5, c=6) | Δ = 1 | Δ = (−5)²−4(1)(6) = 25−24 = 1 > 0. Two distinct real roots: x = 3 and x = 2. |
| x² − 4x + 4 = 0 (a=1, b=−4, c=4) | Δ = 0 | Δ = (−4)²−4(1)(4) = 16−16 = 0. One repeated root: x = 2. The parabola touches the x-axis exactly once. |
| x² + 2x + 5 = 0 (a=1, b=2, c=5) | Δ = −16 | Δ = 4−4(1)(5) = 4−20 = −16 < 0. Two complex conjugate roots: x = −1 ± 2i. The parabola does not cross the x-axis. |
| 2x² − 8x + 6 = 0 (a=2, b=−8, c=6) | Δ = 16 | Δ = 64−4(2)(6) = 64−48 = 16 > 0. Two distinct real roots: x = 3 and x = 1. |
How to Use the Discriminant Calculator
- Identify the coefficients a, b, and c in your quadratic equation written in standard form ax² + bx + c = 0.
- Enter a in the first field, b in the second, and c in the third. Remember a must be non-zero.
- Click Calculate Discriminant to see Δ = b² − 4ac, the root nature, and the roots themselves.
- Use the quick-load buttons to try the three classic examples covering positive, zero, and negative discriminants.
- Click Reset to restore the default values and start a new calculation.
Discriminant Calculator FAQ
What is the discriminant of a quadratic equation?
The discriminant of ax² + bx + c = 0 is the expression Δ = b² − 4ac. It appears under the square root in the quadratic formula and determines the number and type of roots without needing to fully solve the equation. A positive discriminant means two distinct real roots, zero means one repeated root, and negative means two complex conjugate roots.
How do I use the discriminant to find roots?
Once you know Δ, substitute back into the quadratic formula: x = (−b ± √Δ) / (2a). If Δ > 0, take both +√Δ and −√Δ to get two real roots. If Δ = 0, the single root is −b/(2a). If Δ < 0, the roots are complex: x = −b/(2a) ± i√(|Δ|)/(2a).
What does it mean when the discriminant equals zero?
A discriminant of zero means the quadratic has a repeated root (also called a double root). Geometrically, the parabola y = ax² + bx + c is tangent to the x-axis — it just touches it at the vertex without crossing. This happens, for instance, with the perfect square trinomial x² − 4x + 4 = (x−2)².
Can the discriminant be negative?
Yes. A negative discriminant means there is no real square root of Δ, so the quadratic has no real roots. Instead it has two complex conjugate roots of the form p + qi and p − qi. This occurs when the parabola lies entirely above or below the x-axis and never intersects it.
Why does coefficient a have to be non-zero?
If a = 0, the equation ax² + bx + c = 0 reduces to bx + c = 0, which is linear, not quadratic. The quadratic formula and the discriminant are undefined for a = 0 because the denominator 2a would be zero. The calculator requires a ≠ 0 to ensure a genuine quadratic is being analyzed.
How does the discriminant relate to the graph of the quadratic?
The x-intercepts of the parabola y = ax² + bx + c correspond exactly to the real roots of the equation. If Δ > 0, the parabola crosses the x-axis at two distinct points. If Δ = 0, it is tangent to the x-axis at one point (the vertex). If Δ < 0, the parabola does not touch the x-axis at all, confirming that all roots are complex.