A quadratic equation is a mathematical expression that can be used to describe a parabola. It consists of two main components, an x term and a y term, which are combined using one of three possible forms: ax^2 + bx = c, ax^2 + bx + c = 0 or dx^2 + ey^2 = f. The coefficients in the equation represent different aspects of the parabola such as its vertex, direction and orientation. Solving a quadratic equation involves finding the values of x that make the equation true, which can be done using factoring, completing the square, or by using the quadratic formula.