A quadratic algebra is a graded algebra with generators of degree 1 and homogeneous relations of degree 2. The author investigates more general algebras, namely QLS-algebras (quadratic-linear-scalar algebras) and QL-algebras (quadratic-linear algebras) defined by generators and nonhomogeneous relations of degree 2 resp. nonhomogeneous relations of degree 2 without scalar terms. The author generalizes the classical quadratic duality to the above “nonhomogeneous quadratic algebras”. The dual object for a QL-algebra is a quadratic DG-algebra (differential graded algebra) and the dual object for a QLS-algebra is a quadratic CDG-algebra (curved differential graded algebra). A CDG-algebra \(\Psi\) is a triple \((B,d,h)\), where \(B\) is a graded algebra, \(d\) is a derivation on \(B\) of degree 1, and \(h\in B^2\), for which \(d^2=[h,\cdot]\) and \(dh=0\). The classical Poincaré-Birkhoff-Witt theorem can be easily formulated in this duality language. The author presents a generalization of this theorem saying that under the duality every Koszul CDG-algebra corresponds to a QLS-algebra. Further, we can find here examples of these algebras arising in differential geometry. The algebra of differential operators on a manifold may be considered as a QL-algebra defined by commutation relations for vector fields. The dual object is the de-Rham-complex. The algebra of differential operators on a vector bundle is a QLS-algebra. The dual object is the algebra of differential forms with coefficients in linear endomorphisms of this bundle and with differential defined by means of a connection. Finally, the problem of existence of a QL-algebra structure on a QLS-algebra has led the author to obstructions which generalize the Chern classes of vector bundles and Chern-Weil classes of principal bundles. A generalization of the Chern-Simons classes is also presented.