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Yesterday someone asked a question in SE about indeterminate quadratic equations(of the form $x^2−ny^2=1$ which got me really interested in them and I thought I would try to learn something related to it. One of the interesting topics that I found was Pell's equation which led me to the Chakravala method.

I was wondering what are the possible applications of these type of quadratics in real life?

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    $\begingroup$ In "real life"? Like when going to buy bread and milk, or when trying to decide what shoes to buy or what girl to marry? $\endgroup$ – DonAntonio Mar 4 '14 at 12:06
  • $\begingroup$ The secretary problem might be applicable when deciding which girl to marry. There must be some quadratic equation in its solution. :-) $\endgroup$ – MGA Mar 4 '14 at 12:07
  • $\begingroup$ Seriously though, I remember when I was around 14 I came across a set of "real life" problems in my maths textbook whose solutions all required quadratic equations. If I had that textbook handy I would copy them on here for you, but I live in a different country now, and that book's back home. $\endgroup$ – MGA Mar 4 '14 at 12:10
  • $\begingroup$ AFAIU OP isn't asking about quadratic equations as such (there are plenty of examples of their use, any situation that involves constant acceleration leads to them), but about diophantine equations like Pell's. $\endgroup$ – vonbrand Mar 4 '14 at 12:26
  • $\begingroup$ This is a cut-and-paste duplicate question, the original being <math.stackexchange.com/questions/617109/…>. $\endgroup$ – Kieren MacMillan Sep 27 '14 at 13:55
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If your real life involves higher mathematics, then there are plenty of applications of Pell's equation. For one thing, it's where you get units in real quadratic fields. Also, it's where you get good rational approximations to $\sqrt n$.

A lot of puzzle questions come down to Pell equations as well, things like finding right-angle triangles with integer sides with the two legs differing by 1.

I bet if you do a search for Pell or Pellian at this website you'll find dozens of questions where the answer involved solving a Pellian.

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