MHB Infinitely Many Pairs: Integer Roots of Quadratic Equations

[juantheron]

how can we show that there are infinitely many pairs $(a,b)$ such that both the quadratic equations -

$x^2 + ax +b = 0$ and $x^2 +2ax +b = 0$ have integer roots

 

[Nono713]

Re: infinity many pairs

I'll get you started. What is an integer root? A quadratic equation x^2 + ax + b = 0 yields integer roots when \frac{-a \pm \sqrt{a^2 - 4b}}{2} is an integer, with a, b \in \mathbb{Z} (a and b don't need to be relative numbers but if we can prove there are infinitely many pairs of relative numbers \left (a, b \right ) we don't need to worry about non-integers).

Clearly if \frac{-a \pm \sqrt{a^2 - 4b}}{2} is an integer, then -a \pm \sqrt{a^2 - 4b} is one too.

Can you follow this reasoning to the end and establish a condition on a and b such that the resulting quadratic has integer roots? Then, do the same for the other equation with 2a instead and put all the conditions together. Then, prove that all these conditions are satisfied for infinitely many pairs \left (a, b \right ) and you will be done. Does that make sense?