# Quadratic Equations

A quadratic equation is a second degree polynomial equation in two variables. These equations are used in the field of mathematics, specifically in the field of algebra, where they are called quadratic equations. The word "quadratic" has been used since ancient times, at least since 1546. It comes from the Latin "quadratum", which means square. It is a quadratic equation containing one or more squared terms. The equation and its solutions are easily explained using the familiar example of the house and the two men. The quadratic equation is usually written as ax2 + bx + c = 0. The coefficient of x in the numerator is called the "term" of the equation, and the coefficient of x in the denominator is called the "coefficient" of the equation. An example of a quadratic equation is x2 + 2x - 7 = 0. The quadratic equation ax 2 + bx + c = 0 has two solutions, a and b, if and only if the quadratic equation has two real roots, x1 and x2. The two solutions can be found using a process called the process of completing the square. If the quadratic equation has a single real root, then it is called a "real" quadratic equation. If the quadratic equation has more than one real root, it is called an "irrational" quadratic equation. An example of a real quadratic equation is x2 + 2x - 7 = 0. The quadratic equation ax2 + bx + c = 0 has two solutions, a and b, if and only if the quadratic equation has one root, x1, and one real root, x2. The two solutions can be found using a process called the process of factorization. If the quadratic equation has no real roots, it is called a "complex" quadratic equation. An example of a complex quadratic equation is 2x2 - 6x - 9 = 0. When a quadratic equation has more than one solution, there are several methods for finding them. One method is to try them all to see if any work. For example, to find the solutions of 2x2 - 6x - 9 = 0, when 2x + 2 = 0, you try x = -2, -1, 0, 1, and 2. None of these work, so you can try again starting with x = 2. This time, the solutions are x = -3 and x = 1. Other methods are usually faster. There are exactly 4 solvable quadratic equations: x2 - 1 = 0, x2 + 1 = 0, x2 - x = 0, and x2 + x = 0. There are exactly 3 solvable real quadratic equations: x2 - 1 = 0, x2 + 1 = 0, and x2 - x = 0. There are exactly 2 solvable complex quadratic equations: x2 - 1 = 0, and x2 + x = 0. If the quadratic equation has a real root, it is real. If the quadratic equation has a complex root, it is complex. If the quadratic equation has no real roots, it is complex.