Problem
ALG-B1-M09-P012 Quadratic form equals 7
#12
★★★☆☆ Level 3 of 5
Find all integer pairs \((x,y)\) such that \(x^2+xy+y^2=7\).
Use \(4(x^2+xy+y^2)=(2x+y)^2+3y^2\).
We get \((2x+y)^2+3y^2=28\). Thus \(3y^2\le28\), so \(y\in\{-3,-2,-1,0,1,2,3\}\). Checking these seven values gives all solutions: \((-3,1),(-3,2),(-2,-1),(-2,3),(-1,-2),(-1,3),(1,-3),(1,2),(2,-3),(2,1),(3,-2),(3,-1)\).
Bounding a variable through a positive quadratic form.