State equations that are at most quadratic with respect to the states are considered as a universal model structure for various classes of nonlinear systems. A necessary and sufficient condition is shown for existence of such quadratic-in-the-state representation for a given nonlinear system. Furthermore, the constructive proof of that necessary and sufficient condition gives an algorithm to obtain the quadratic-in-the-state representation. Illustrative examples demonstrate the algorithm to construct the quadratic-in-the-state representation. The quadratic-in-the-state representation is expected to be useful in identification and control system design.