Solve ax² + bx + c = 0 instantly — real or complex roots, discriminant, vertex, and axis of symmetry.
Enter the three coefficients of your equation in the standard form ax² + bx + c = 0. The calculator applies the quadratic formula and returns both roots, whether they're real or complex, along with the discriminant, vertex coordinates, and axis of symmetry for the corresponding parabola.
The roots of ax² + bx + c = 0 are given by x = (−b ± √(b² − 4ac)) / 2a. The expression under the square root, b² − 4ac, is called the discriminant, and its sign determines what kind of roots the equation has.
A positive discriminant means the equation has two distinct real roots, where the parabola crosses the x-axis at two points. A discriminant of exactly zero means there is one repeated real root, where the parabola's vertex sits exactly on the x-axis. A negative discriminant means the equation has two complex conjugate roots, meaning the parabola never crosses the x-axis at all.
Complex roots occur when the discriminant is negative, which happens when the parabola doesn't intersect the x-axis. The roots still exist mathematically as a pair of complex conjugates in the form p ± qi, but they don't correspond to real x-intercepts on a standard graph.
The vertex is the turning point of the parabola — its minimum if the parabola opens upward, or maximum if it opens downward. Its x-coordinate is h = −b / 2a, and the y-coordinate k is found by substituting h back into the original equation.
If a = 0, the equation is no longer quadratic — it becomes a linear equation, bx + c = 0, which has at most one root rather than two. This calculator requires a non-zero value for a to solve as a genuine quadratic.