  Let $k$ be a field and let $\text{GW}(k)$ be the Grothendieck-Witt ring of
virtual non-degenerate symmetric bilinear forms over $k$. We develop methods
for computing the quadratic Euler characteristic $\chi(X/k)\in \text{GW}(k)$
for $X$ a smooth hypersurface in a projective space or a weighted projective
space. We raise the question of a quadratic refinement of classical conductor
formulas and find such a formula for the degeneration of a smooth hypersurface
$X$ in $\mathbb{P}^{n+1}$ to the cone over a smooth hyperplane section of $X$;
we also find a similar formula in the weighted homogeneous case. We formulate a
conjecture that generalizes these computations to similar types of
degenerations. Finally, we give an interpretation of the quadratic conductor
formulas in terms of Ayoub's nearby cycles functor.
