Quadratic Formula
Introduction
The quadratic formula is a solution to the quadratic equation, a fundamental concept in algebra. The quadratic formula provides a method to solve all quadratic equations, including those that are not factorable and those that do not have rational roots.
Definition
The quadratic formula is defined as follows:
If a quadratic equation is in the form ax^2 + bx + c = 0, where a, b, and c are constants, then the solutions to the equation, often referred to as the roots of the equation, can be found using the quadratic formula:
x = [-b ± sqrt(b^2 - 4ac)] / (2a)
The term b^2 - 4ac is known as the discriminant, and it provides crucial information about the roots of the quadratic equation.
Derivation
The quadratic formula is derived from the process of completing the square on a quadratic equation. This method involves adding a constant to both sides of the equation to create a perfect square trinomial on one side of the equation.
Applications
The quadratic formula is widely used in many fields of study, including physics, engineering, and finance. It is used to solve problems involving parabolic motion, circuit design, and investment analysis, among others.
Properties
The quadratic formula has several important properties. For instance, if the discriminant is positive, the quadratic equation has two distinct real roots. If the discriminant is zero, the equation has one real root (also known as a repeated root). If the discriminant is negative, the equation has two complex roots.