Quadratic Formula Calculator
Solve quadratic equations and visualize parabolas with step-by-step solutions
Quadratic Equation: ax² + bx + c = 0
Two Real Roots
Root 1: x₁ = 3.0000
Root 2: x₂ = 2.0000
Discriminant: Δ = 1.0000
Vertex: (2.5000, -0.2500)
Y-intercept: 6
Axis of symmetry: x = 2.5000
Opens: Upward
What is Quadratic Formula Calculator?
What is the Quadratic Formula?
The quadratic formula is used to solve quadratic equations of the form ax² + bx + c = 0, where a ≠ 0. The formula is: x = (-b ± √(b² - 4ac)) / (2a)
The Discriminant
The discriminant (Δ = b² - 4ac) determines the nature of the roots:
- Δ > 0: Two real and distinct roots
- Δ = 0: One real root (repeated)
- Δ < 0: Two complex conjugate roots
Key Components
- Vertex: The turning point of the parabola at x = -b/(2a)
- Axis of Symmetry: The vertical line x = -b/(2a)
- Y-intercept: The point where the parabola crosses the y-axis (0, c)
- Direction: Opens upward if a > 0, downward if a < 0
Applications
Quadratic equations appear in many real-world scenarios including projectile motion, optimization problems, area calculations, and physics problems involving acceleration.
FAQ - Quadratic Formula Calculator
If a = 0, the equation becomes linear (bx + c = 0) and is no longer quadratic. The quadratic formula only applies when a ≠ 0.