The quadratic equation ax² + bx + c = 0 appears everywhere: physics (projectile motion), engineering (beam deflection), finance (break-even analysis), and computer graphics (ray-sphere intersection). The quadratic formula x = (-b ± √(b²-4ac)) / 2a provides the solution for any coefficients.
How to Use
Enter the coefficients a, b, and c:
- a: Coefficient of x² (must not be 0)
- b: Coefficient of x
- c: Constant term
The Discriminant
The discriminant D = b² - 4ac determines the nature of roots:
- D > 0: Two distinct real roots
- D = 0: One repeated real root
- D < 0: Two complex conjugate roots
Vertex Form
Every quadratic can be written as a(x - h)² + k where (h, k) is the vertex. h = -b/(2a) is the axis of symmetry, and k = c - b²/(4a) is the minimum (or maximum) value.
Step-by-Step Instructions
- 1Enter coefficient a (x² term).
- 2Enter coefficient b (x term).
- 3Enter coefficient c (constant).
- 4View the roots and discriminant analysis.
- 5Study the step-by-step solution.