  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.
