The paper concerns global solvability of initial value problem for one class of hyperbolic quasilinear second order equations with two independent variables, which have a rather wide range of applications. Besides existence and uniqueness of maximal solutions of this problem it is proved that a maximal solution possess the completeness property that is an analog of the corresponding property of ordinary differential equations. Namely,a solution of an ordinary differential equation that is defined on a maximal interval leaves any compact subset of the equation domain.