For a quadratic polynomial with rational coefficients, we consider the problem of
bounding the number of rational points that eventually land at a given constant after
iteration, called preimages of the constant. It was shown by Faber, Hutz, Ingram, Jones,
Manes, Tucker, and Zieve (2009) that the number of rational preimages is bounded as
one varies the polynomial. Explicit bounds on the number of preimages of zero and
were
addressed in subsequent articles. This article addresses explicit bounds on
the number of preimages of any algebraic number for quadratic dynamical
systems and provides insight into the geometric surfaces parameterizing such
preimages.