Problem
NT-B2-M07-P009 Shifting a Square
#9
★★★☆☆ Level 3 of 5
Find all positive integer pairs \((x,y)\) such that \(x^2+4x=y^2+11\).
Complete \(x^2+4x\) to the square \((x+2)^2\).
We have \((x+2)^2-y^2=15\), so \((x+2-y)(x+2+y)=15\). The positive factor pairs with the same parity are \((1,15)\) and \((3,5)\), both odd. They give \((x,y)=(6,7)\) and \((2,1)\). Both check in the original equation.
This trains the passage from a quadratic expression to a difference of squares.