Problem
ALG-B1-M09-P016 Discriminant as a square
#16
★★★☆☆ Level 3 of 5
Find all integers \(k\) for which \(x^2-6x+k=0\) has integer roots.
The roots have sum \(6\). List integer pairs with that sum.
Let the roots be \(r\) and \(s\). Then \(r+s=6\), \(k=rs\). Write \(r=3+t\), \(s=3-t\), where \(t\in\mathbb Z\). Then \(k=9-t^2\). Conversely, for every integer \(t\), the roots \(3+t\) and \(3-t\) are integers. Answer: all \(k=9-t^2\), \(t\in\mathbb Z\).
Parametrizing through the sum of roots is often easier than using the discriminant.