Quadratic Equation Calculator
Enter the coefficients a, b, and c of ax² + bx + c = 0 to find its roots and discriminant.
a·x² + b·x + c = 0
How the quadratic formula works
For any equation in the form ax² + bx + c = 0 (with a ≠ 0), the roots are given by the quadratic formula: x = (−b ± √(b² − 4ac)) / (2a).
The discriminant, b² − 4ac, determines the type of roots. If it's positive, there are two distinct real roots. If it's zero, there's exactly one real root (a repeated root). If it's negative, there are two complex conjugate roots involving the imaginary unit i.
Frequently asked questions
What does the discriminant tell you?
The discriminant (b² − 4ac) tells you how many real roots the equation has: positive means two real roots, zero means one repeated real root, and negative means two complex roots.
What happens if a = 0?
If a = 0, the equation is no longer quadratic — it becomes linear (bx + c = 0) and the quadratic formula doesn't apply, so this calculator requires a to be nonzero.
What are complex roots?
When the discriminant is negative, the square root of a negative number isn't a real number, so the roots are expressed using the imaginary unit i (where i² = −1) as a pair of complex conjugates.