Quadratic families of elliptic curves and unirationality of degree 1 conic bundles

We study families of elliptic curves whose coefficients are degree 2 polynomials in a variable t. All such curves together form an algebraic surface which is birational to a conic bundle with 7 singular fibers. We prove that such conic bundles are unirational. As a consequence one obtains that, for infinitely many values of t, the resulting elliptic curve has rank at least 1.