Solve ax² + bx + c = 0 and see the discriminant, roots, vertex, y-intercept, and a parabola graph.
Quadratic equation solver
Step-by-step solution
| Discriminant | Roots | Graph meets x-axis |
|---|---|---|
| D > 0 | Two distinct real roots | Two points |
| D = 0 | One real root (double) | One point (tangent) |
| D < 0 | Two complex roots (p ± qi) | No intersection |
What does the discriminant tell me?
The discriminant D = b² − 4ac describes the nature of the roots. D > 0: two distinct real roots. D = 0: one repeated real root. D < 0: two complex conjugate roots p ± qi.
How do I read complex roots?
When D < 0 the roots are written as p ± qi, where p = −b/(2a) is the real part and q = √|D|/(2a) is the imaginary part.
What is the vertex and how is it found?
The vertex (h, k) is the turning point: h = −b/(2a), k = (4ac − b²)/(4a). The line x = h is the axis of symmetry.
What happens if a = 0?
The equation becomes linear. This calculator requires a ≠ 0 and shows an error for a = 0.
This quadratic formula calculator solves any equation of the form ax² + bx + c = 0. Enter the three coefficients and the tool returns the discriminant D = b² − 4ac, classifies the nature of the roots, and computes both solutions using x = (−b ± √D) / (2a). When D is negative, the calculator writes the complex conjugate pair p ± qi instead of showing an error. It also reports the vertex (h, k) with h = −b/(2a) and k = (4ac − b²)/(4a), the axis of symmetry, the y-intercept, and the direction in which the parabola opens. A step-by-step panel shows the formula substitution line by line, and an SVG plot marks the roots, vertex, and y-intercept so you can sanity-check the result visually. Example: for x² − 5x + 6 = 0 the discriminant is 1, and the roots are x₁ = 3 and x₂ = 2.