A quadratic equation has the standard form ax² + bx + c = 0, where a ≠ 0. The graph of a quadratic is a parabola — a U-shaped curve that opens upward when a > 0 and downward when a < 0.
There are three main methods to solve quadratic equations. Factoring works when the equation can be written as (px + q)(rx + s) = 0. The quadratic formula, x = (-b ± √(b²-4ac)) / 2a, works for any quadratic. Completing the square transforms the equation into vertex form.
The discriminant, Δ = b² - 4ac, determines the nature of the solutions. When Δ > 0, there are two distinct real solutions. When Δ = 0, there is exactly one real solution (a repeated root). When Δ < 0, there are no real solutions — instead, there are two complex solutions involving imaginary numbers.
Quadratic equations appear in projectile motion, area optimization problems, and revenue/profit modeling in business.