On the surface, this talk is about a continued fraction expansion of the function Γ(x+3/4)²/Γ(x+1/4)². However, since this function is rather artificial, the actual topic will be the techniques used to prove the result, i.e. the Euler-Maclaurin formula, formal power series and functional equation trickery. As a generalization, we can show that, given any positive-valued quadratic polynomial P, the continued fraction x + P(1)/(x+ P(2)/(x+ ...)) converges for all x > 0 and can be characterized via a functional equation.