In this paper the class of mixed Horn formulas is introduced that contain a Horn part and a 2-CNF (conjunctive normal form) (also called quadratic) part. We show that SAT remains NP-complete for such instances and also that any CNF formula can be encoded in terms of a mixed Horn formula in polynomial time. Further, we provide an exact deterministic algorithm showing that SAT for mixed Horn formulas containing n variables is solvable in time O ( 2 0.5284 n ) . A strong argument showing that it is hard to improve a time bound of O ( 2 n / 2 ) for mixed Horn formulas is provided. We also obtain a fixed-parameter tractability classification for SAT restricted to mixed Horn formulas C of at most k variables in its positive 2-CNF part providing the bound O ( ∥ C ∥ 2 0.5284 k ) . We further show that the NP-hard optimization problem minimum weight SAT for mixed Horn formulas can be solved in time O ( 2 0.5284 n ) if non-negative weights are assigned to the variables. Motivating examples for mixed Horn formulas are level graph formulas [B. Randerath, E. Speckenmeyer, E. Boros, P. Hammer, A. Kogan, K. Makino, B. Simeone, O. Cepek, A satisfiability formulation of problems on level graphs, ENDM 9 (2001)] and graph colorability formulas.