The Discriminant The quadratic formula, , can be a useful tool in determining the solution(s) to a quadratic equation.  The nature of the solution(s) can be determined by a quick look at the discriminant.  The radicand of the quadratic formula is the discriminant.  In other words the discriminant is b2 - 4ac. Using the standard form of the quadratic equation which is , the nature of the solution(s) to any quadratic formula can then be determined by checking the discriminant. (1) If b2 - 4ac = 0, then the quadratic equation will have exactly one Real number solution (with duplicity.)  In other words, there is one solution that is repeated.  This occurs in the case of a perfect square trinomial such as x2 - 6x + 9 whose discriminant would be (-6)2 - 4(1)(9) = 0.  When factored you would normally arrive at the solution (x -3)(x -3) = 0 or x = 3 and x =3, thus the "duplicity" of the answer x = 3. (2) If b2 - 4ac > 0 (in other words the discriminant is a positive number), then the quadratic equation will have exactly two Real number Solutions.  For example consider, x2 - 2x -8 whose discriminant would be (-2)2 - 4(1)(-8) = 36.  Since 36 is greater than 0, there are two (unequal) Real solutions. (3) If b2 - 4ac < 0 (in other words the discriminant is a negative number), then the quadratic equation will have no Real solutions.  Instead there will be two Complex number solutions. For example consider, x2 - 3x + 6 whose discriminant would be (-3)2 - 4(1)(6) = -15.  Since -15 is less than 0, there are two Complex Number (conjugates) solutions.   General Algebra Tips The views and opinions expressed in this page are strictly those of Mary Lou Baker. The contents of this page have not been reviewed or approved by Columbia State Community College.This page was edited on 20-Jan-2011