ax² + bx + c = 0 the discriminant is D = b² − 4ac. Two real roots when D > 0, a repeated root when D = 0, a conjugate pair when D < 0: x = (−b ± √D) / (2a). The vertex is at x = −b / (2a). If a is 0 the equation is not quadratic.
For any equation in the form ax² + bx + c = 0, the discriminant D = b² − 4ac determines what kind of roots the equation has before you even solve it — positive, zero, or negative each tell a different story about the solution.
When D is greater than zero, the equation has two distinct real roots. When D equals zero, there's exactly one repeated real root. When D is negative, the two roots are a complex conjugate pair — the parabola never crosses the x-axis at all. In every case, the roots are x = (−b ± √D) / (2a).
The vertex of the parabola — its lowest or highest point — sits at x = −b / (2a), which is also the axis of symmetry. If a is 0, the equation collapses to a straight line rather than a parabola, so it isn't a quadratic at all and needs a linear solver instead.
Its sign predicts the type of roots: positive means two distinct real roots, zero means one repeated real root, and negative means a complex conjugate pair with no real solutions — useful for understanding the shape of the parabola before computing exact values.
The calculator still shows them as a conjugate pair in the form p ± qi, using the same formula with a negative value under the square root handled as an imaginary number.
The vertex is the parabola's turning point, at x = −b / (2a) — it's the minimum point if a is positive, or the maximum point if a is negative, and it's also the parabola's axis of symmetry.
The equation is no longer quadratic (there's no x² term), so this calculator won't produce a valid quadratic solution — you'd need a linear equation solver instead for bx + c = 0.