  We develop a method for solving the field equations of a quadratic
gravitational theory coupled to matter. The quadratic terms are written as a
function of the matter stress tensor and its derivatives in such a way to have,
order by order, a set of Einstein field equations with an effective
$T_{\mu\nu}$. We study the cosmological scenario recovering the de Sitter exact
solution, and the first order (in the coupling constants $\alpha$ and $\beta$
appearing in the gravitational Lagrangian) solution to the gauge cosmic string
metric and the charged black hole. For this last solution we discuss the
consequences on the thermodynamics of black holes, and in particular, the
entropy - area relation which gets additional terms to the usual ${1\over4} A$
value.
