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We present a projection-based nonlinear model order reduction method, named model order reduction via quadratic-linear systems (QLMOR). QLMOR employs two novel ideas: 1) we show that nonlinear ordinary differential equations, and more generally differential-algebraic equations (DAEs) with many commonly encountered nonlinear kernels can be rewritten equivalently in a special representation, quadratic-linear differential algebraic equations (QLDAEs), and 2) we perform a Volterra analysis to derive the Volterra kernels, and we adapt the moment-matching reduction technique of nonlinear model order reduction method (NORM)  to reduce these QLDAEs into QLDAEs of much smaller size. Because of the generality of the QLDAE representation, QLMOR has significantly broader applicability than Taylor-expansion-based methods - since there is no approximation involved in the transformation from original DAEs to QLDAEs. Because the reduced model has only quadratic nonlinearities, its computational complexity is less than that of similar prior methods. In addition, QLMOR, unlike NORM, totally avoids explicit moment calculations, hence it has improved numerical stability properties as well. We compare QLMOR against prior methods - on a circuit and a biochemical reaction-like system, and demonstrate that QLMOR-reduced models retain accuracy over a significantly wider range of excitation than Taylor-expansion-based methods -. QLMOR, therefore, demonstrates that Volterra-kernel based nonlinear MOR techniques can in fact have far broader applicability than previously suspected, possibly being competitive with trajectory-based methods (e.g., trajectory piece-wise linear reduced order modeling ) and nonlinear-projection based methods (e.g., maniMOR ).