  We prove a simple necessary and sufficient condition for a two-bridge knot
K(p,q) to be quasipositive, based on the continued fraction expansion of p/q.
As an application, coupled with some classification results in contact and
symplectic topology, we give a new proof of the fact that smoothly slice
two-bridge knots are non-quasipositive. Another proof of this fact using
methods within the scope of knot theory is presented in the Appendix.
