The problem of the robust and optimal control for uncertain quadratic systems is dealt with in this paper. Resorting to a guaranteed cost approach, this paper proposes a novel control design methodology which enables to find a state feedback controller guaranteeing for the closed-loop system: i) the local asymptotic stability of the zero equilibrium point; ii) the inclusion of a given polytopic region into the domain of attraction of the zero equilibrium point; iii) the satisfaction of a quadratic performance index. The control performance is guaranteed against parametric uncertainties which are assumed to be norm-bounded. This design procedure involves the solution of a Linear Matrix Inequalities (LMIs) optimization problem, which can be efficiently solved via off-the-shelf algorithms. An example, concerning an application of motion control for robotic arms, shows the effectiveness of the proposed methodology.