Solving Quadratic Equations

A quadratic equation is any equation that can be rearranged in standard form as ax² + bx + c = 0, where a, b, and c are known coefficients and x is the unknown variable. Quadratic equations appear throughout science, engineering, economics, and everyday life — from calculating projectile trajectories to optimizing profit margins.

1. The Quadratic Formula

The quadratic formula provides the exact solutions to any quadratic equation: x = (-b ± √(b² - 4ac)) / 2a. This formula works for all quadratic equations, whether the roots are real or complex. The term under the square root, b² - 4ac, is called the discriminant and determines the nature of the solutions.

2. The Discriminant

The discriminant (Δ = b² - 4ac) reveals the nature of the roots without solving the equation: (1) If Δ > 0, there are two distinct real roots — the parabola crosses the x-axis at two points. (2) If Δ = 0, there is exactly one real root (a repeated root) — the parabola touches the x-axis at its vertex. (3) If Δ < 0, there are no real roots — the parabola does not intersect the x-axis, and the solutions are complex numbers.

3. The Vertex and Graph

The vertex of the parabola y = ax² + bx + c is at the point (h, k) where h = -b/(2a) and k = f(h). The vertex represents the maximum value of the function if a < 0 (parabola opens downward) or the minimum value if a > 0 (parabola opens upward). The axis of symmetry is the vertical line x = h, which divides the parabola into two mirror-image halves.