“A beautiful little problem!”

I’ve been going through my old Twitter saves, when I came across this one which was no longer available, but which intrigued me!

The post is now gone from Twitter, but I found it via the WaybackMachine.

Prove that if all the coefficients of the quadratic equation \[ax^2 + bx + c = 0\] are odd integers, then the roots of the equation cannot be rational.

Want a hint for solving the problem? Highlight the following hidden text:

Write down \(1^2, 3^2, 5^2\) and \(7^2\) mod \(8\).

Use this together with the quadratic formula.

This problem comes fromThe USSR Olympiad Problem Book (1962) by D. O. Shklarsky. The whole book can be found digitally on archive.org.