Solving Quadratic EquationsBy analytic and graphic methods; Including several methods you may never have seen Pat Ballew, 2007I received a copy of an oldcolumn in the Mathematics Teacher (March, 1951, pp 193-194) from David Renfro, who writes those wonderful math questions for the people at the ACT, and also regularly takes time out of his busy schedule to educate me about topics I have overlooked, under covered, or just plain done wrong on my math words web page. In this case the column reminded me of the old song, Fifty Ways to Leave Your Lover,50 Ways To Leave Your LoverThe problem is all inside your headShe said to meThe answer is easy if youTake it logicallyI’d like to help you in your struggle To be freeThere must be fifty waysTo leave your loverIn this column, however, the logic was applied tosolving quadratics, and unfortunately, there are only 18(ok, I’ll add a couple not in the list below, so maybe I will show 20 ways to solve a quadratic equation). I suppose if the song had come out earlier, Professor Hazard, whose letter is a major topic of the column, could have sliced, diced and stretched the existing 18, into a few more, but I think the 18 (plus or minus a few)that are here will serve as a learning experience for most teachers, and almost any student. And speaking of songs, I am reminded of the importance and prestige

that comes with understanding quadratics in yet another song. In Gilbert and Sullivan's operetta The Pirates of Penzance, Major General Stanley proudly proclaims to the pirates his knowledge of quadratic equations, among other skills, in "The Major General's Song". "I am the very model of a modern Major-General, I've information vegetable, animal, and mineral, I know the kings of England, and I quote the fights historical, From Marathon to Waterloo, in order categorical; I'm very well acquainted too with matters mathematical, Iunderstand equations, both the simple and quadratical, About binomial theorem I'm teeming with a lot o' news-- With many cheerful facts about the square of the hypotenuse."I will copy Dave’s Note that has part of thecolumn that lists the eighteen waysProfessor Hazard presents, and then for each method I’ll try to give an example, anda little explanation, a historical note where I think it is helpful, and perhaps an extension where I think it is needed.Dave’s Note saidWhat follows is part of the Mathematical Miscellanea column edited by Phillip S. Jones in MathematicsTeacher 44 #3 (March 1951), pp. 193-194 (pages for this part only, not the entire column)..A letter from Willian [sic?] J. Hazard of the Department of Engineering Mathematics of the University of Colorado includes the following list of 18 ways to solve ax^2 + bx + c = 0 taken from an article which he published in January 1924 in the 'Colorado Engineer':1. By factoring by inspection.