The discriminant Δ = b²−4ac determines the nature of roots: Δ>0 → two distinct real roots; Δ=0 → double root; Δ<0 → no real roots.
Quadratic Equation Solver
Solve ax²+bx+c=0 with the quadratic formula, discriminant and real or complex roots.
Quadratic formula with discriminant and step-by-step roots.
The quadratic equation ax²+bx+c=0 (a≠0) is solved by the quadratic formula: x = (−b ± √Δ) / 2a, where Δ = b²−4ac. If Δ>0 there are two distinct real roots; if Δ=0, a double root; if Δ<0, the roots are complex conjugates with no real solution.
Enter coefficients a, b and c.
- Type coefficient a (must be ≠ 0).
- Type coefficients b and c (can be zero).
- See discriminant Δ, root type and values x₁ and x₂.
Sources and references for this tool
These references help contextualize formulas, standards, APIs and limitations used on this page. They do not replace professional validation when a result has legal, financial, medical or operational impact.
- Quadratic Formula — Wolfram MathWorldWolfram MathWorld — Quadratic formula, derivation and discriminant cases.
- Quadratic Equation — WikipediaWikipedia — History, forms and solution methods for quadratic equations.
- Solving Quadratics — Khan AcademyKhan Academy — Interactive review of the quadratic formula with examples.