Problem
ALG-B1-M09-P010 Difference of squares with coefficient
#10
★★☆☆☆ Level 2 of 5
Find all integer solutions of \(x^2-4y^2=12\).
Factor as \((x-2y)(x+2y)=12\).
Let \(u=x-2y\), \(v=x+2y\). Then \(uv=12\), \(x=\frac{u+v}{2}\), \(y=\frac{v-u}{4}\). We need \(u+v\) divisible by \(2\), and \(v-u\) by \(4\). Checking divisors of \(12\) gives \((u,v)=(2,6),(6,2),(-2,-6),(-6,-2)\). Hence \((x,y)=(4,1),(4,-1),(-4,-1),(-4,1)\).
Trains parity conditions after factorization.