The aim of this paper is to study what happens when a slight perturbation affects the coefficients of a quadratic equation defining a variety (a quadric) in R^n. Structurally stable quadrics are those a small perturbation on the coefficients of the equation defining them does not give rise to a "different" (in some sense) set of points. In particular we characterize structurally stable quadrics and give the "bifurcation diagrams" of the non stable ones (showing which quadrics meet all of their neighbourhoods), when dealing with the "affine" and "metric" equivalence relations. This study can be applied to the case where a set of points which constitute the set of solutions of a problem is deffined by a quadratic equation whose coefficients are given with parameter uncertainty.