Enter the coefficients a, b, and c of a quadratic equation in the form ax² + bx + c = 0 to calculate the discriminant (D = b² − 4ac). The Discriminant Calculator returns the discriminant value and tells you whether the roots are two distinct real roots, one repeated real root, or two complex (non-real) roots. Also try the find Standard Form Equation with Equation of a Sphere Calculator.
Discriminant (D)
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Nature of Roots
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Root 1 (x₁)
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Root 2 (x₂)
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The discriminant is the part of the quadratic formula under the square root sign, expressed as D = b² − 4ac for the equation ax² + bx + c = 0. It tells you how many real solutions the equation has and whether those solutions are rational or irrational. See also our Radical Calculator.
The discriminant reveals the nature of a quadratic equation's roots without fully solving it. A positive discriminant means two distinct real roots, zero means exactly one repeated real root, and a negative discriminant means two complex (non-real) roots. This is crucial in algebra, physics, and engineering.
If D > 0, the equation has two distinct real roots. If D = 0, there is exactly one repeated (equal) real root. If D < 0, the equation has two complex conjugate roots with no real solutions. Additionally, if D is a perfect square and a, b, c are rational, the roots are rational.
For a quadratic equation ax² + bx + c = 0, the discriminant is D = b² − 4ac. It forms the expression inside the square root in the quadratic formula x = (−b ± √D) / (2a). You might also find our calculate Adding and Subtracting Polynomials Result useful.
No. If a = 0, the equation is no longer quadratic — it becomes a linear equation (bx + c = 0). The discriminant formula and quadratic formula both require a ≠ 0 to be valid.
When D = 0, the quadratic equation has exactly one real root, also called a repeated or double root. The root is given by x = −b / (2a). Geometrically, the parabola touches the x-axis at exactly one point (the vertex).
When D < 0, the square root of a negative number produces imaginary values. The two roots become complex conjugates: x = (−b ± i√|D|) / (2a), where i is the imaginary unit (√−1). These roots cannot be plotted on a real number line.
Yes, discriminants exist for higher-degree polynomials, but the formula is different. For a cubic ax³ + bx² + cx + d, the discriminant is Δ = 18abcd − 4b³d + b²c² − 4ac³ − 27a²d². This calculator focuses specifically on the quadratic case using D = b² − 4ac.