A study of flash equations associated with multicomponent systems that can form three phases at a given temperature and pressure indicated that different equations are needed at different boundaries of the solution space. This problem dictates that one- and two-phase trial solutions be obtained first on the boundaries prior to a search in the three-phase interior region. Two new formulations of the flash equations have been developed to avoid this problem. Solutions at boundaries as well as those in the three-phase region can now be obtained with a single search procedure. The new formulations are made using functions that are monotonic in the selected coordinates. These properties facilitate the use of nested, single-variable search methods. Interpolating with second-order functions, when applicable, nested search methods are often faster than two-dimensional simultaneous search methods. Tests of these formulations have demonstrated the advantages of these characteristics.