Quadratic Equation Calculator — Roots, Steps & Graph
Solve quadratic equations in standard or vertex form, or build one from roots. See real or complex roots, discriminant, vertex, steps, forms, points, and a graph.
Solve ax² + bx + c = 0
Enter the three coefficients to find real or complex roots and analyze the parabola.
Parabola graph
Equation analysis
| Standard | |
| Vertex | |
| Factored | |
| Axis | |
| y-intercept |
Step-by-step solution
Key points near the vertex
| Point | x | y = ax² + bx + c |
|---|
Quadratic formula calculator
For ax² + bx + c = 0, the roots are x = (−b ± √(b² − 4ac)) / 2a. This solver calculates the discriminant first and returns two real roots, one repeated root, or a complex conjugate pair.
Vertex and axis of symmetry
The vertex lies at h = −b / 2a and k = f(h). The vertical line x = h is the axis of symmetry; the sign of a determines whether the parabola opens upward or downward.
Standard, vertex, and factored forms
Enter coefficients, start with a(x − h)² + k, or build an equation from two known roots. The calculator converts every input to standard form and shows the equivalent forms that exist.
Complex quadratic roots
When the discriminant is negative, the graph has no real x-intercepts, but the equation still has two complex roots. They are displayed as a conjugate pair using the imaginary unit i.
