Quadratic Equation Solver
Solve ax² + bx + c = 0 for its roots — real or complex — using the quadratic formula, with the discriminant, axis of symmetry and the parabola’s vertex.
ax² + bx + c = 0
Roots
x₁ = 2, x₂ = 1
Two distinct real roots
Discriminant (b² − 4ac)
1
Nature of Roots
Two distinct real roots
Axis of Symmetry
x = 1.5
Vertex
(1.5, -0.25)
How It Works
A quadratic equation has the form ax² + bx + c = 0 (with a ≠ 0), and the quadratic formula solves it: x equals negative b plus or minus the square root of (b² − 4ac), all divided by 2a. The expression under the root, b² − 4ac, is the discriminant, and it tells you the nature of the roots before you compute them: if it is positive there are two distinct real roots, if zero there is one repeated real root, and if negative there are two complex-conjugate roots. Beyond the roots, the calculator also reports the axis of symmetry (x = −b/2a) and the vertex — the turning point of the parabola. Enter the coefficients a, b and c (they can be negative or decimals) and it shows the roots and all the key features of the parabola instantly.
Formula
x = (−b ± √(b² − 4ac)) ÷ 2a. Discriminant D = b² − 4ac: D>0 two real roots, D=0 one repeated, D<0 two complex.
Frequently Asked Questions
What is the quadratic formula?
x = (−b ± √(b² − 4ac)) ÷ 2a. It gives both roots of ax² + bx + c = 0 for any a ≠ 0, whether the roots are real or complex.
What does the discriminant tell me?
The discriminant b² − 4ac reveals the roots’ nature: positive means two distinct real roots, zero means one repeated real root, and negative means two complex-conjugate roots.
What if a = 0?
Then it is not a quadratic but a linear equation (bx + c = 0), and the quadratic formula does not apply. The calculator flags this — enter a non-zero value for a.
Can it handle complex roots?
Yes. When the discriminant is negative, the calculator returns the two complex-conjugate roots in the form p ± qi rather than leaving the equation unsolved.
What is the vertex of a parabola?
The turning point of the curve y = ax² + bx + c, located at x = −b/2a. It is the minimum (if a > 0) or maximum (if a < 0) of the parabola, and the calculator reports it.