Quadratic Equation Solver

Analyzing parabolic motion and optimization curves requires solving second-degree polynomial equations across physics, engineering, and economics. Users input three numerical coefficients labeled a, b, and c corresponding to the standard algebraic form ax² + bx + c = 0. The leading coefficient a defines whether the parabola opens upward or downward, while the linear coefficient b and constant term c determine the vertical placement and horizontal axis of symmetry across the coordinate plane.

The generated output presents the equation roots alongside the calculated discriminant and vertex coordinates. A positive discriminant signifies two distinct real roots where the curve intersects the horizontal axis, a zero discriminant denotes a single repeated real root, and a negative discriminant indicates complex conjugate roots containing imaginary components. Note that the leading coefficient a must not be zero, as setting it to zero collapses the quadratic expression into a standard linear equation.

The solver evaluates the discriminant Δ = b² - 4ac to establish root multiplicity and determine whether solutions are real or complex. Applying the quadratic formula x = (-b ± √Δ) / (2a) yields the exact root values. Parabola vertex coordinates are calculated using h = -b / (2a) for the horizontal axis of symmetry and evaluating the polynomial at h to find the vertical extreme.