2010/018) The set of all integers n for which sqrt(n^2 + n) is an integer is...?

The set of all integers n for which sqrt(n^2 + n) is an integer is...?
a) the set {0,-1};
b) a finite set with at least three elements;
c) an infinite set;
d) none of these sets;
Kindly explain...

Ans:
sqrt(n^2 + n) = sqrt(n(n+1)) is inetger

n and n+1 are coprimes so either n= 0 or n = - 1 or n and n+1 both squares

n = x^2 and n+1 = y^2 => 1 = (x+y)(y-x) => x+y = 1 and y-x = 1 => x = 0 y =1 => n = 0

or x+y = -1 and y-x = -1 => x = 0 y =-1 => n = 0

so only solution 0 and -1 so ans is a)