The Quadratic Formula
x = [−b ± √(b² − 4ac)] ÷ 2a
This formula solves any equation in the standard form ax² + bx + c = 0, where a, b, and c are known coefficients and a ≠ 0.
The Discriminant Determines the Type of Solution
The expression under the square root, b² − 4ac, is called the discriminant and determines the nature of the solutions:
- Positive discriminant: Two distinct real solutions
- Zero discriminant: One repeated real solution (the parabola touches the x-axis at exactly one point)
- Negative discriminant: Two complex (imaginary) solutions — the parabola never crosses the x-axis
Common Applications
- Algebra and pre-calculus coursework
- Physics problems involving projectile motion (finding time when height equals zero)
- Engineering and optimization problems involving quadratic relationships