Problem
NT-B1-M05-P016 A Square with a Linear Term
#16
★★★☆☆ Level 3 of 5
Find all positive integer solutions of \(x^2+2x=y^2+20\).
Add \(1\) to the left-hand side to obtain \((x+1)^2-y^2\).
Rewrite as \((x+1)^2-y^2=21\). Then \((x+1-y)(x+1+y)=21\). The positive factor pairs are \((1,21)\) and \((3,7)\). They give respectively \((x,y)=(10,10)\) and \((4,2)\). There are no other solutions.
A useful example where factorisation appears after completing a square.