A Ternary Algebra with Applications to Binary Quadratic Forms

We discuss multiplicative properties of the binary quadratic form $a x^2 + b x y + c y^2$ by considering a ring of matrices which is closed under a triple product. We prove that the ring forms a ternary algebra in the sense of Hestenes, and then derive both multiplicative formulas for a large class of binary quadratic forms and a type of multiplication for points on a conic section which generalizes the algebra of rational points on the unit circle. Published as part of the proceedings for the Council for African American Researchers in the Mathematical Sciences (CAARMS)