Dual basis sets are employed as an economical way to approximate self-consistent field (SCF) calculations, such as Kohn-Sham density functional theory (DFT), in large basis sets. First, an SCF calculation is performed in a small subset of the full set of basis functions. The density matrix in this small basis is used to construct the effective Hamiltonian operator in the large basis, from which a correction for basis set extension is obtained for the energy. This correction is equivalent to a single Roothaan step (diagonalization) in the large basis. We present second order nonlinear equations that permit this step to be obtained without explicit diagonalization. Numerical tests on part of the Gaussian-2 dataset, using the B3LYP density functional, show that large-basis results can be accurately approximated with this procedure, subject to some limitations on the smallness of the small basis. Computational savings are approximately an order of magnitude relative to a self-consistent DFT calculation in the large basis.