  We study elliptic surfaces over $\mathbb{Q}(T)$ with coefficients of a
Weierstrass model being polynomials in $\mathbb{Q}[T]$ with degree at most 2.
We derive an explicit expression for their rank over $\mathbb{Q}(T)$ depending
on the factorization and other simple properties of certain polynomials.
Finally, we give sharp estimates for the ranks of the considered families and
we present several applications, among which there are lists of rational
points, generic families with maximal rank and generalizations of former
results.
