The maximum 2-satisfiability problem (MAX-2-SAT) is: given a Boolean formula in 2-CNF, find a truth assignment that satisfies the maximum possible number of its clauses. MAX-2-SAT is MAX-SNP-complete. Recently, this problem received much attention in the contexts of (polynomial-time) approximation algorithms and (exponential-time) exact algorithms. In this paper, we present an exact algorithm solving MAX-2-SAT in time poly(L) · 2^K/5, where K is the number of clauses and L is their total length. In fact, the running time is only poly(L) · 2^K2/5, where K₂ is the number of clauses containing two literals. This bound implies the bound poly(L) · 2^L/10. Our results significantly improve previous bounds: poly(L) ·2 ^K/2.88 (J. Algorithms 36 (2000) 62-88) and poly(L) · 2^K/3.44 (implicit in Bansal and Raman (Proceedings of the 10th Annual Conference on Algorithms and Computation, ISAAC'99, Lecture Notes in Computer Science, Vol. 1741, Springer, Berlin, 1999, pp. 247-258.)). As an application, we derive upper bounds for the (MAX-SNP-complete) maximum cut problem (MAX-CUT), showing that it can be solved in time poly(M) · 2^M/3, where M is the number of edges in the graph. This is of special interest for graphs with low vertex degree.