This paper is devoted to analytical and numerical investigation of limit cycles in two-dimensional polynomial quadratic systems. The appearance of modern computers permits one to use a numerical simulation of complicated nonlinear dynamical systems and to obtain new information on a structure of their trajectories. However the possibilities of naive approach, based on the construction of trajectories by numerical integration of the considered differential equations, turns out to be very limited. In the paper the effective analytical-numerical methods for investigation of limit cycles in two-dimensional polynomial quadratic system are discussed. Estimations of domains of parameters, corresponding to existence of different configurations of large limit cycles, are obtained and visualization of four large limit cycles in quadratic system is presented.