Consider the equation x2+2x−n=0 where n∈N and n∈[5,100]. The total number of different values of n so that the given equation has integral roots is

x2+2x−n=0⇒(x+1)2=n+1⇒ x=−1±n+1Thus, n + 1 should be a perfect square. Now,n∈[5,100]⇒n+1∈[6,101]Perfect square values of n + 1 are 9,16,25,36,49,64,81, 100.Hence, number of values is 8.