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This work addresses the design of state feedback controllers for locally stabilizing open-loop unstable quadratic systems with guaranteed stability domain and performance. First, a method is derived to design a stabilizing nonlinear static state feedback controller, which is quadratic in the system state, while providing an enlarged stability region for the closed-loop system. The stabilization method is then extended in three directions. The first one is to ensure a quadratic regulator-type performance, the second is the local stabilization, in sense of integral input-to-state stability, of quadratic systems disturbed by energy-bounded exogenous signals, whereas the third extension provides a solution to the H∞ control problem. The developed control methods are tailored via finite sets of state-dependent linear matrix inequalities. Several numerical examples are presented to illustrate the potentials of the proposed controller designs.