Polynomial, geometric, binomial, and factorial sequences all satisfy first-order recurrence relations and lie within the class of rational-valued hypergeometric sequences. The Threshold Problem for hypergeometric sequences is as follows: given such a sequence (u_n)_n and a rational constant t, decide whether u_n ≥ t for each n. We shall discuss challenging aspects of the Threshold Problem in this setting and how recent work establishes (un)conditional decidability results for subclasses of hypergeometric sequences---those with quadratic parameters alluded to in the title.