  Matrix congruence can be used to mimic linear maps between homogeneous
quadratic polynomials in $n$ variables. We introduce a generalization, called
standard-form congruence, which mimics affine maps between non-homogeneous
quadratic polynomials. Canonical forms under standard-form congruence for
three-by-three matrices are derived. This is then used to give a classification
of algebras defined by two generators and one degree two relation. We also
apply standard-form congruence to classify homogenizations of these algebras.
