Problem
ALG-B1-M10-P018 Consecutive roots
#18
★★★★☆ Level 4 of 5
Find all integer pairs \((a,b)\) for which \(x^2+ax+b\) has two integer roots differing by \(1\).
Let the roots be \(n\) and \(n+1\).
Let the roots be \(n\) and \(n+1\), where \(n\in\mathbb Z\). By Vieta, \(a=-(2n+1)\), \(b=n(n+1)\). Conversely, for such \(a,b\), the polynomial is \((x-n)(x-n-1)\). Answer: \((a,b)=(-(2n+1),n(n+1))\), \(n\in\mathbb Z\).
Parametrization instead of search.