Understanding the Quadratic Formula
The quadratic formula is a universal method for solving quadratic equations of the form ax² + bx + c = 0, where a, b, and c are real numbers and a is not equal to zero. The formula gives both solutions directly.
The Discriminant
The discriminant D = b² - 4ac determines the nature of the roots:
D > 0: Two Real Roots
The parabola crosses the x-axis at two distinct points.
x1 != x2, both real
D = 0: One Repeated Root
The parabola touches the x-axis at exactly one point (vertex).
x1 = x2 = -b / 2a
D < 0: Two Complex Roots
The parabola does not cross the x-axis. Roots are complex conjugates.
x = (-b +/- i*sqrt(|D|)) / 2a
Vertex and Axis of Symmetry
Every quadratic equation y = ax² + bx + c describes a parabola. The vertex is the highest or lowest point, and the axis of symmetry passes through the vertex vertically.
- Axis of symmetry: x = -b / (2a)
- Vertex: (-b/(2a), f(-b/(2a)))
- Opens upward if a > 0, downward if a < 0.
Alternative Methods
- Factoring: Works when roots are rational. Factor ax² + bx + c = a(x - r1)(x - r2).
- Completing the square: Rewrite in vertex form a(x - h)² + k = 0.
- Graphing: Plot y = ax² + bx + c and find x-intercepts visually.
Vieta's Formulas
For a quadratic ax² + bx + c = 0 with roots r1 and r2:
- Sum of roots: r1 + r2 = -b/a
- Product of roots: r1 * r2 = c/a