square method, represent interesting applications of other frequently used techniques in algebra, or offer insight into other areas of mathematics. A lesser known quadratic formula, as used in Muller's method provides the same roots via the equation This can be deduced from the standard quadratic formula by Vieta's formulas, which assert that the product of the roots is . It also follows from dividing the quadratic equation by giving solving this for and then inverting. One property of this form is that it yields one valid root when , while the other root contains division by zero, because when , the quadratic equation becomes a linear equation, which has one root. By contrast, in this case, the more common formula has a division by zero for one root and an indeterminate form for the other root. On the other hand, when , the more common formula yields two correct roots whereas this form yields the zero root and an indeterminate form . When neither nor is zero, the equality between the standard quadratic formula and Muller's method, can be verified by cross multiplication, and similarly for the other choice of signs. Reduced quadratic equation It is sometimes convenient to reduce a quadratic equation so that its leading coefficient is one. This is done by dividing both sides by , which is always possible since is non-zero. This produces the reduced quadratic equation: where and . This monic polynomial equation has the same solutions as the original. The quadratic formula for the solutions of the reduced quadratic equation, written in terms of its coefficients, is: or equivalently: Discriminant In the quadratic formula, the expression underneath the square root sign is called the discriminant of the quadratic equation, and is often represented using an upper case or an upper case Greek delta: A quadratic equation with real coefficients can have either one or two distinct real roots, or two distinct complex roots. In this case the discriminant determines the number and nature of the roots. There are three cases: If the discriminant is positive, then there are two distinct roots both of which are real numbers. For quadratic equations with rational coefficients, if the discriminant is a square number, then the roots are rational—in other cases they may be quadratic irrationals. If the discriminant is zero, then there is exactly one real root sometimes called a repeated or double root. If the