Quadratic Equation Solver

Solve quadratic equations (ax² + bx + c = 0). Find roots, discriminant, vertex, and axis of symmetry with a visual graph.

This calculator solves ax² + bx + c = 0 using the quadratic formula (line 66-67), and first computes the discriminant b² − 4ac (line 60) to decide which of three branches applies: two distinct real roots when the discriminant is positive, one repeated root when it's exactly 0, or two complex conjugate roots when it's negative (lines 65-79). At the defaults (a=1, b=−5, c=6), b dominates the discriminant's sensitivity — because b enters as b², moving b further from 0 in either direction always increases the discriminant, while moving it closer to 0 decreases it; a and c, by contrast, each enter linearly through the −4ac term, so raising either one (holding the other fixed and positive) always lowers the discriminant. The discriminant actually crosses zero — the boundary between real and complex roots — at b ≈ −4.899 for these values of a and c, since b² = 4ac = 24 there. The Vertex and Axis of Symmetry (lines 82-83) are computed the same way regardless of the discriminant's sign, since −b/2a is defined whenever a ≠ 0. This calculator does not handle a = 0: that branch (lines 8-58) silently falls back to solving the linear equation bx + c = 0 instead, and reports Root 2, Discriminant, Vertex, and Axis of Symmetry as "N/A" rather than computing them, since a linear equation has no parabola to describe.