  Inference of the equation-of-state (EoS) of dense nuclear matter in
neutron-star cores is a principal science goal of X-ray and gravitational-wave
observations of neutron stars. In particular, gravitational-wave observations
provide an independent probe of the properties of bulk matter in neutron star
cores that can then be used to compare with theoretically derived equations of
state. In this paper, we quantify the systematic errors arising from the
application of EoS-independent \emph{quasi-universal relations} in the
estimation of neutron star tidal deformabilities and radii from
gravitational-wave measurements and introduce a strategy to correct for the
systematic biases in the inferred radii. We apply this method to a simulated
population of events expected to be observed by future upgrades of current
detectors and the next-generation of ground-based observatories. We show that
our approach can accurately correct for the systematic biases arising from
approximate universal relations in the mass-radius curves of neutron stars.
Using the posterior distributions of the mass and radius for the simulated
population we infer the underlying EoS with a good degree of precision. Our
method revives the possibility of using the universal relations for rapid
Bayesian model selection of dense matter EoS in gravitational-wave
observations.
