Anecdotes please - quadratic equations in real life

[Allen]

useful for factoring certain special form numbers.

guess near 854 (any number from 854-859 will factor n using the formula below),
by converting n to base 854 then solve quadratic.

\(
\begin{align}
n= 859*997 \\

guess = 854 \\
n = (1,148,715) _{854} \\
(a,b,c) = (1,148,715) \\

g - \frac{-b + \sqrt{b^2 -4ac}} {2} = 859\\
g - \frac{-b - \sqrt{b^2 -4ac}} {2} = 997\\

\end{align}
\)

Even if your guess is elsewhere, you can make fairly large steps (knowing the number has no factor in that range) by solving the quadratic equation there on the number in base g.

For example, guessing 883, the quadratic eq returns 894, then guessing 894 it returns 914... it's not practical for many circumstances, but this method works well when \( n \) has a factor that's around \( \sqrt n - a \sqrt {\sqrt{n}} \) for some "small" a.