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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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.