  Parity-violating electron scattering provides a model-independent
determination of the nuclear weak-charge form factor that has widespread
implications across such diverse areas as fundamental symmetries, nuclear
structure, heavy-ion collisions, and neutron-star structure. We assess the
impact of precise measurements of the weak-charge form factor of ${}^{48}$Ca
and ${}^{208}$Pb on a variety of nuclear observables, such as the neutron skin
and the electric-dipole polarizability. We use the nuclear Density Functional
Theory with several accurately calibrated non-relativistic and relativistic
energy density functionals. To assess the degree of correlation between nuclear
observables and to explore systematic and statistical uncertainties on
theoretical predictions, we employ the chi-square statistical covariance
technique. We find a strong correlation between the weak-charge form factor and
the neutron radius, that allows for an accurate determination of the neutron
skin of neutron-rich nuclei. We determine the optimal range of the momentum
transfer $q$ that maximizes the information content of the measured weak-charge
form factor and quantify the uncertainties associated with the strange quark
contribution. Moreover, we confirm the role of the electric-dipole
polarizability as a strong isovector indicator. Accurate measurements of the
weak-charge form factor of ${}^{48}$Ca and ${}^{208}$Pb will have a profound
impact on many aspects of nuclear theory and hadronic measurements of neutron
skins of exotic nuclei at radioactive-beam facilities.
