Quadratic Equation Solver

The quadratic formula solves ax² + bx + c = 0: x = (−b ± √(b²−4ac)) / 2a. The ± gives two solutions — x₁ uses addition, x₂ uses subtraction.

The discriminant is b² − 4ac. If positive: two distinct real roots. If zero: one repeated real root. If negative: two complex (imaginary) roots. The discriminant tells you the nature of solutions without fully solving the equation.

The vertex is the highest or lowest point of the parabola y = ax² + bx + c. Its x-coordinate is −b/(2a), and the y-coordinate is c − b²/(4a). The vertex is also where the axis of symmetry passes.

When the discriminant is negative, the square root produces an imaginary number. Complex roots come in conjugate pairs: x = p ± qi, where i = √(−1). These roots don't appear on the real number line but are valid solutions in the complex plane.