Quadratic Equation Solver
A quadratic equation has the standard form ax² + bx + c = 0, where a ≠ 0. This solver finds both roots using the quadratic formula, classifies the nature of the roots via the discriminant, and calculates the parabola's vertex.
Quadratic Formula
The Discriminant (Δ = b² − 4ac)
The discriminant determines how many real solutions exist:
- Δ > 0: Two distinct real roots. The parabola crosses the x-axis at two points.
- Δ = 0: One repeated real root (double root). The parabola just touches the x-axis.
- Δ < 0: Two complex conjugate roots. The parabola does not cross the x-axis. Roots take the form p ± qi.
Vertex of the Parabola
Vertex y = c − b² / 4a
The vertex is the turning point of the parabola. If a > 0, the vertex is the minimum point. If a < 0, it is the maximum.
Example
Solve x² − 5x + 6 = 0 (a = 1, b = −5, c = 6):
x = (5 ± 1) / 2
x₁ = 3, x₂ = 2
Frequently Asked Questions
Can a be zero?
No. If a = 0, the x² term disappears and the equation becomes linear (bx + c = 0), not quadratic. The solver will reject a = 0.
What are complex roots used for?
Complex roots appear in electrical engineering (impedance), signal processing, control systems, and quantum mechanics. In a school context, they indicate the parabola has no real x-intercepts.