  A weaker form of a 1979 conjecture of Goldfeld states that for every elliptic
curve $E/\mathbb{Q}$, a positive proportion of its quadratic twists $E^{(d)}$
have rank 1. Using tools from Galois cohomology, we give criteria on E and d
which force a positive proportion of the quadratic twists of E to have 3-Selmer
rank 1 and global root number -1. We then give four nonisomorphic infinite
families of elliptic curves $E_{m, n}$ which satisfy these criteria.
Conditional on the rank part of the Birch and Swinnerton-Dyer conjecture, this
verifies the aforementioned conjecture for infinitely many elliptic curves. Our
elliptic curves are easy to give explicitly and we state precisely which
quadratic twists d to use. Furthermore, our methods have the potential of being
generalized to elliptic curves over other number fields.
