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?