Quadratic Equation Solver
Solve ax² + bx + c = 0 — real or complex roots, a graph of the parabola, vertex form, and the full working.
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How to solve a quadratic equation
- Write your equation as ax² + bx + c = 0 and enter a, b and c.
- Read the roots, vertex form and the graph of the parabola.
- Click Show step-by-step working for the full method.
About this tool
The discriminant D = b² − 4ac determines the root type: positive gives two distinct real roots, zero gives one repeated real root (the parabola just touches the x-axis at its vertex), and negative gives a complex conjugate pair rather than an error — the parabola simply never crosses the x-axis, which the graph makes visible directly.
Vertex form a(x − h)² + k is found by completing the square, with h = −b ⁄ 2a and k = c − b² ⁄ 4a. The vertex is the parabola's turning point — a minimum when a > 0, a maximum when a < 0 — and its x-coordinate h is also the axis of symmetry and the average of the two roots when they're real.
Nothing you enter here is sent to a server — the entire tool, including the graph, runs as JavaScript in your browser tab.
Questions about quadratic equations
What is the quadratic formula?
x = (−b ± √(b² − 4ac)) ÷ 2a. It solves any equation of the form ax² + bx + c = 0.
What does the discriminant tell me?
b² − 4ac: positive means two real roots, zero means one repeated root, and negative means two complex roots.
What if there are no real roots?
The parabola doesn’t cross the x-axis, and the roots are a complex pair, which the solver shows.
How do I find the vertex?
Its x-coordinate is −b ÷ 2a; put that back into the equation for y. The solver gives the vertex form too.