We verify a conjecture of Emil Artin, for the case of a Cubic and Quadratic
form over any $p$-adic field, provided the cardinality of the residue class
field exceeds 293. That is any Cubic and Quadratic form with at least 14
variables has a non-trivial $p$-adic zero, with the aforementioned condition on
the residue class field.

A crucial step in the proof, involves generalizing a $p$-adic minimization
procedure due to W. M. Schmidt to hold for systems of forms of arbitrary
degrees.