If a polynomial equation is solved for multiple roots in a system of equilibria (e.g. calculating extent of a reaction, or solving for [H+] in a complicated acid-base system, to give two basic examples) how do we know which, among the roots, is the correct solution (e.g. the correct extent or the correct proton concentration for the two cases above)? Is it always (in analytical chemistry/equilibrium situations) the smallest real, positive root which is the one we should take? We can assume the root which correctly represents the solution must be real and positive (and often, smaller than a certain upper limit we can impose, as e.g. in the case of extent) but must it be the smallest, and if not, how do we choose the correct root?