The control of smooth input-affine nonlinear systems can be performed using exact linearization techniques which are based on the existence of a fictitious output map producing a (nonlinear) relative degree equal to the system order. Although the existence of such a function is easily determinded its computation is usually quite difficult. In this paper a method is proposed how to explicitly construct a map having the desired properties for a special class of nonlinear systems. This class is obtained by taking into account only the linear and quadratic part of the drift vector and the constant and linear part of the input vector of a general nonlinear system and can be thought of as a better approximation of that system as the usual linear or bilinear approaches are. One important fact presented in this paper is that under some additional assumptions on the "higher-order" terms more general results are obtainable then those obtained by the well-known "matching conditions".