Going beyond the Golden Ratio

They agree to play a game where Emily thinks of a number, and then Sam (with the possible help of his minions) has 60 seconds to find any fractions that are equal to Emily’s number. But for any other smaller score $S$, (even infinitesimally smaller than this one), Sam and his Minions would only be able to find a finite number of fractions with a score less than $S$. Emily continued her exploration of quadratic irrationals and soon learnt than whenever she picked a number that was an irrational root of a quadratic equation $ax^2+bx+c=0$ for integers $a,b,c$, the critical lower threshold would involve (if not be identical) to the expression $ 1/\sqrt{b^2-4ac}$.

Source: extremelearning.com.au