A quadratic equation consists of a single variable with three terms in the standard form: ax2 + bx + c = 0. The first quadratic equations were developed as a method used by Babylonian mathematicians around 2000 BC to solve simultaneous equations. Quadratic equations can be applied to problems in physics involving parabolic motion, path, shape, and stability. Several methods have evolved to simplify the solution of such equations for the variable x. Any number of quadratic equation solvers, in which the values of the quadratic equation coefficients can be entered and automatically calculated, can be found online.
The three methods most commonly used to solve quadratic equations are factoring, completing the square, and the quadratic formula. Factoring is the simplest form of solving a quadratic equation. When the quadratic equation is in its standard form, it is easy to visualize if the constants a, b, and c are such that the equation represents a perfect square. First, the standard form must be divided through by a. Then, half of, what is now, the b/a term must be equal to twice, what is now, the c/a term; if this is true, then the standard form can be factored into the perfect square of (x ± d)2.
If the solution of a quadratic equation is not a perfect square and the equation cannot be factored in its present form, then a second solution method — completing the square — can be used. After dividing through by the a term, the b/a term is divided by two, squared, and then added to both sides of the equation. The square root of the perfect square can be equated to the square root of all the remaining constants on the right hand side of the equation in order to find x.
The final method of solving the standard quadratic equation is by directly substituting the constant coefficients (a, b, and c) into the quadratic formula: x = (-b±sqrt(b2-4ac))/2a, which was derived by the method of completing the squares in the generalized equation. The discriminant of the quadratic formula (b2 - 4ac) appears under a square root sign and, even before the equation is solved for x, can indicate the type and number of solutions found. The type of solution depends on whether the discriminant is equal to the square root of a positive or negative number. When the discriminant is zero, there is only one positive root. When the discriminant is positive, there are two positive roots, and when the discriminant is negative, there are both positive and negative roots.