Keynote Address: Quadratic Equations: Can We Really Apply Them?,Victor Katz

In a first course in school algebra, one of the culminating ideas is the quadratic formula for the solution of quadratic equations.  But why is this so important?  Are there quadratic equations out there in the real world that we need to solve?  Mohammad al-Khwarizmi, back in the ninth century, thought that quadratic equations were important, so important that their solutions, using what we might consider versions of the quadratic formula, were at the center of his own algebra text, the first true textbook in algebra.  Later Islamic authors extended al-Khwarizmi’s work, and it was this Islamic work that was transmitted to Europe in the late Middle Ages and became the core of Europe’s own work in algebra.  Did Islamic mathematicians or medieval and Renaissance European mathematicians find important problems to solve that required the knowledge of the quadratic formula?  We will take a quick tour of algebra textbooks in various countries, from the ninth century through the seventeenth century, to see why mathematicians continued to stress the importance of solving quadratic equations.  We will then look at the implications of this importance in the teaching of algebra today.