Quadratic Equation Solver
Solve ax² + bx + c = 0, including complex roots, vertex and discriminant.
What the Quadratic Equation Solver does
The quadratic formula solves any equation of the form ax² + bx + c = 0. The discriminant b² − 4ac decides the outcome: positive gives two real roots, zero gives one repeated root, and negative gives a conjugate pair of complex roots.
Formula
x = (−b ± √(b² − 4ac)) ÷ 2aDiscriminant Δ = b² − 4acVertex at x = −b ÷ 2a
Inputs explained
| Input | Unit | Required | Notes |
|---|---|---|---|
| a (x² coefficient) | number | Yes | — |
| b (x coefficient) | number | Yes | — |
| c (constant) | number | Yes | — |
How to use it
- Enter a (x² coefficient), b (x coefficient) and c (constant).
- Select Calculate.
Worked example
Solve x² − 5x + 6 = 0.
- a
- 1
- b
- -5
- c
- 6
Δ = 25 − 24 = 1, so x = (5 ± 1) ÷ 2, giving roots 3 and 2.
Reading the result
- The discriminant tells you what kind of roots to expect before you see them: positive means two distinct real roots, zero means one repeated real root, and negative means the roots are a complex-conjugate pair — not that the equation has no solution.
Common mistakes
- Reading a negative discriminant as "no answer". The equation still has two roots; they are complex rather than real, and this calculator shows them rather than omitting them.
Frequently asked questions
What does the discriminant tell me?
How many times the parabola crosses the x-axis: twice if positive, once (a tangent touch) if zero, never if negative.
What are complex roots useful for?
They show the parabola never meets the x-axis, and they matter in signal processing, control systems and electrical engineering.