Quadratic Equation Calculator
Solve any quadratic equation of the form ax² + bx + c = 0 using the quadratic formula. This calculator finds both roots (real or complex), computes the discriminant, identifies the root type, and locates the vertex of the parabola.
A quadratic equation is a second-degree polynomial equation in a single variable x, with the general form ax² + bx + c = 0, where a is not equal to zero.
The Quadratic Formula
x = (-b ± sqrt(b² - 4ac)) / (2a)
This formula gives both solutions to any quadratic equation.
The Discriminant (D = b² - 4ac):
The discriminant determines the nature of the roots:
- If D > 0: Two distinct real roots
- If D = 0: One repeated real root (the parabola touches the x-axis at exactly one point)
- If D < 0: Two complex conjugate roots (the parabola does not cross the x-axis)
The Vertex
Every quadratic equation y = ax² + bx + c defines a parabola. The vertex (the highest or lowest point) is located at h = -b/(2a), k = c - b²/(4a). If a > 0, the parabola opens upward and the vertex is a minimum. If a < 0, it opens downward and the vertex is a maximum.
Factoring vs. Quadratic Formula
Simple quadratics can be solved by factoring, but the quadratic formula works for all cases, including those with irrational or complex roots.