Quadratic equation
Definition
A quadratic equation is a second-order polynomial equation in a single variable x, with a nonzero coefficient for x^2. The general form is ax^2+bx+c=0, where x represents an unknown variable, and a, b, and c are constants with a ≠ 0. If a = 0, then the equation is linear, not quadratic.
History
The solution to the quadratic equation was known as early as 2000 BC. The Babylonians could solve certain quadratic equations, such as x^2 = 2. The Greeks, in the form of Euclid's Elements, could solve certain quadratics, such as x^2 = a. The solution to the general quadratic equation was described by the Indian mathematician Brahmagupta in the 7th century AD.
Derivation
The quadratic equation is derived from the process of completing the square in the quadratic expression ax^2 + bx + c. This process transforms the quadratic equation into the form (x-h)^2 = k, which allows for the easy solution of the equation.
Solutions
The solutions to the quadratic equation are given by the quadratic formula: x = [-b ± sqrt(b^2 - 4ac)] / (2a). The term b^2 - 4ac is known as the discriminant and it determines the nature of the roots of the equation. If the discriminant is positive, the equation has two distinct real roots. If it is zero, the equation has one real root (or a repeated root). If it is negative, the equation has two complex roots.
Graphical Representation
The graph of a quadratic equation is a parabola. The vertex of the parabola is the point (h, k), where h = -b/2a and k = c - b^2/4a. The axis of symmetry of the parabola is the line x = h. The parabola opens upwards if a > 0 and downwards if a < 0.
Applications
Quadratic equations arise in a variety of practical applications. For example, they are used in physics to model motion under constant acceleration, in economics to model cost and revenue functions, and in engineering to model stress and strain in materials.