Discriminant Calculator
The discriminant is a mathematical expression that reveals the nature of the roots of a quadratic equation without requiring you to solve it. For any quadratic equation in the standard form ax² + bx + c = 0, the discriminant is calculated as D = b² - 4ac. This single value tells you whether the equation has two distinct real roots, one repeated real root, or two complex conjugate roots. In algebra and pre-calculus, understanding the discriminant is essential for analyzing quadratic functions, determining where parabolas cross the x-axis, and solving systems of equations. The discriminant appears directly in the quadratic formula as the expression under the square root sign, which is why its sign and value determine the type and number of solutions. This calculator computes the discriminant instantly and interprets the result, helping you understand the structure of any quadratic equation.
Discriminant formula
For the quadratic equation: ax² + bx + c = 0
Discriminant: D = b² - 4ac
Interpreting the discriminant
| Discriminant Value | Root Type | Number of Roots | Description |
|---|---|---|---|
| D > 0 | Real and distinct | 2 | Two different real numbers satisfy the equation |
| D = 0 | Real and equal | 1 (repeated) | One real root with multiplicity 2 |
| D < 0 | Complex conjugates | 2 (complex) | Two complex roots of the form a ± bi |
Discriminant calculator: frequently asked questions
What is the discriminant?
The discriminant is the expression b² - 4ac in a quadratic equation ax² + bx + c = 0. It determines the nature of the roots without solving the equation. The discriminant tells you whether the roots are real or complex, and whether they are distinct or repeated.
What does a positive discriminant mean?
When the discriminant is positive (D > 0), the quadratic equation has two distinct real roots. These are two different real numbers that satisfy the equation. The roots are found using the quadratic formula: x = (-b ± sqrt(D)) / (2a).
What does zero discriminant mean?
When the discriminant equals zero (D = 0), the quadratic equation has exactly one repeated real root. This single root has multiplicity 2, meaning it appears twice in the factorization of the quadratic. The root is x = -b / (2a).
What does a negative discriminant mean?
When the discriminant is negative (D < 0), the quadratic equation has two complex conjugate roots. These are not real numbers but involve the imaginary unit i. Complex roots come in pairs: a + bi and a - bi, where a and b are real numbers.
How is the discriminant used in the quadratic formula?
The quadratic formula is x = (-b ± sqrt(D)) / (2a), where D is the discriminant. When D > 0, the square root is real and gives two roots. When D = 0, the square root is zero and gives one repeated root. When D < 0, the square root is imaginary and gives complex roots.
Official sources
- Khan Academy: Quadratic equations and the discriminant.
- Wolfram MathWorld: Discriminant.
Reviewed by the CalculatorHub team, edited by James Graham, 14 June 2026. See our methodology.