Problem
ALG-B1-M02-P017 Integer Solutions After Completing a Square
#17
★★★☆☆ Level 3 of 5
Find all integers \(x,y\) such that \(x^2-y^2=2x+6\).
Move \(2x\) to the left and complete the square.
Rewrite as \(x^2-2x-y^2=6\). Then \((x-1)^2-y^2=7\).
Thus \((x-1-y)(x-1+y)=7\). The possible factor pairs are \((1,7)\), \((7,1)\), \((-1,-7)\), \((-7,-1)\).
Solving the systems gives \((x,y)=(5,3),(5,-3),(-3,-3),(-3,3)\).
Here factorisation appears only after preparing the expression.