Quadratic Formula Calculator
Solve quadratic equations ax² + bx + c = 0 and find roots, discriminant, and vertex.
The Quadratic Formula Calculator solves any equation of the form ax² + bx + c = 0 using the quadratic formula: x = (-b ± √(b²-4ac)) / 2a.
How It Works
Enter the coefficients a, b, and c from your quadratic equation. The calculator computes both roots (x₁ and x₂), the discriminant (b²-4ac), and the vertex of the parabola.
Understanding the Discriminant
- Positive discriminant: Two distinct real roots
- Zero discriminant: One repeated real root (the parabola touches the x-axis)
- Negative discriminant: Two complex conjugate roots (no real x-intercepts)
The vertex is calculated as (-b/2a, f(-b/2a)) and represents the minimum or maximum point of the parabola. This tool is essential for algebra students, engineers, and anyone solving second-degree polynomial equations.
Frequently Asked Questions
What happens when the discriminant is negative?
When b²-4ac < 0, the equation has no real solutions. The roots are complex numbers. The calculator will indicate this with complex root notation.
Can the coefficient 'a' be zero?
No. If a = 0, the equation becomes linear (bx + c = 0), not quadratic. The calculator requires a non-zero value for 'a'.
What is the vertex of the parabola?
The vertex is the highest or lowest point of the parabola, calculated as x = -b/(2a). If a > 0, it's a minimum; if a < 0, it's a maximum.