to which they are exposed. If one is given a quadratic equation in the form , the sought factorization has the form , and one has to find two numbers and that add up to and whose product is (this is sometimes called "Vieta's rule" and is related to Vieta's formulas). As an example, factors as . The more general case where does not equal can require a considerable effort in trial and error guess-and-check, assuming that it can be factored at all by inspection. Except for special cases such as where or , factoring by inspection only works for quadratic equations that have rational roots. This means that the great majority of quadratic equations that arise in practical applications cannot be solved by factoring by inspection. Completing the square The process of completing the square makes use of the algebraic identity which represents a well-defined algorithm that can be used to solve any quadratic equation. Starting with a quadratic equation in standard form, Divide each side by , the coefficient of the squared term. Subtract the constant term from both sides. Add the square of one-half of , the coefficient of , to both sides. This "completes the square", converting the left side into a perfect square. Write the left side as a square and simplify the right side if necessary. Produce two linear equations by equating the square root of the left side with the positive and negative square roots of the right side. Solve each of the two linear equations. We illustrate use of this algorithm by solving The plus–minus symbol "±" indicates that both and are solutions of the quadratic equation. Quadratic formula and its derivation Completing the square can be used to derive a general formula for solving quadratic equations, called the quadratic formula. The mathematical proof will now be briefly summarized. It can easily be seen, by polynomial expansion, that the following equation is equivalent to the quadratic equation: Taking the square root of both sides, and isolating , gives: Some sources, particularly older ones, use alternative parameterizations of the quadratic equation such as or , where has a magnitude one half of the more common one, possibly with opposite sign. These result in slightly different forms for the solution, but are otherwise equivalent. A number of alternative derivations can be found in the literature. These proofs are simpler than the standard completing the