Quadratic Formula Calculator
Enter coefficients (a, b, c) from a quadratic polynomial to instantly find real or complex roots, the discriminant, and vertex coordinates.
Quadratic Target: ax² + bx + c = 0
Roots (One Real Root)
x = -2
Discriminant (Δ)
0
Vertex (x, y)
(-2, 0)
Understanding Output of the Quadratic Formula
The quadratic formula is a universal mathematical tool used to find the roots (or x-intercepts) of any quadratic equation formatted as ax² + bx + c = 0. It elegantly solves equations that cannot be easily factored, providing exact intersection points where a parabola crosses the x-axis.
The Discriminant (Δ)
The expression underneath the square root symbol in the quadratic formula, known as the discriminant (b² - 4ac), directly informs you about the nature of the roots without having to solve the entire equation:
- Positive Discriminant (Δ > 0): The equation has two distinct real roots. The parabola crosses the x-axis at exactly two different points.
- Zero Discriminant (Δ = 0): The equation has exactly one real root (a repeated root). The vertex of the parabola rests exactly on the x-axis, touching it at a single point.
- Negative Discriminant (Δ < 0): The equation has two complex (imaginary) roots. The parabola is floating entirely above or below the x-axis and never explicitly intersects it in the real plane.
The Vertex
The calculator also provides the exact (x, y) coordinate of the parabola's vertex. The vertex is the absolute highest or lowest point of the curve. The x-coordinate of the vertex is found at -b / (2a), lying exactly halfway between the two roots along the axis of symmetry.