  Let K be a field of any characteristic and let R be an algebra generated by
two elements satisfying quadratic equations. Then R is a homomorphic image of
F=K<x,y | x^2+ax+b=0,y^2+cy+d=0> for suitable a,b,c,d in K. We establish that F
can be embedded into the 2x2 matrix algebra M_2(E[t]) with entries from the
polynomial algebra E[t] over the algebraic closure E of K and that F and M_2(E)
satisfy the same polynomial identities as K-algebras. When the quadratic
equations have double zeros, our result is a partial case of more general
results by Ufnarovskij, Borisenko and Belov from the 1980's. When each of the
equations has different zeros, we improve a result of Weiss, also from the
1980's.
