In this paper, we are dealing with the solution of the functional equation ϕ x + y 2 (f (x) − f (y)) = F(x) − F(y), concerning the unknown functions ϕ, f and F defined on a same open subinterval of the reals. Improving the previous results related to this topic, we describe the solution triplets (ϕ, f ,F) assuming only the continuity of ϕ. As an application, under natural conditions, we also solve the equality problem of twovariable Cauchy means and two-variable quasi-arithmetic means.