Quadratic Equation Solver

A quadratic equation is a polynomial equation of degree 2, written in the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. The graph of a quadratic equation is a parabola, which is a U-shaped curve that opens upward if a > 0 or downward if a < 0.

The quadratic formula is x = (-b ± √(b² - 4ac)) / (2a). It provides a direct method to find the solutions (roots) of any quadratic equation. The ± symbol means there are typically two solutions - one using addition and one using subtraction.

The discriminant is the expression b² - 4ac inside the quadratic formula. It determines the nature of the roots: if discriminant > 0, there are two distinct real roots; if discriminant = 0, there is exactly one real root (a repeated root); if discriminant < 0, there are two complex conjugate roots.

The vertex of a parabola given by y = ax² + bx + c is at the point (h, k) where h = -b/(2a) and k = c - b²/(4a). The vertex represents the maximum point (if a < 0) or minimum point (if a > 0) of the parabola.

Yes! When the discriminant is negative, the quadratic equation has two complex conjugate roots. This solver calculates and displays them in the form a + bi and a - bi, where i = √(-1). The graph will show that the parabola does not cross the x-axis in this case.

Vertex form is y = a(x - h)² + k, where (h, k) is the vertex of the parabola. It's useful because you can immediately identify the vertex coordinates, determine whether the parabola opens up (a > 0) or down (a < 0), and see the maximum or minimum value of the function (k).