Let the consecutive positive integers be x and x+1.
Therefore, x2+(x+1)2=365
⇒x2+x2+1+2x=365
⇒2x2+2x−364=0
⇒x2+x−182=0
(Splitting the middle term x as 14x-13x because 14x×−13x=−182x2)
⇒x2+14x−13x−182=0
⇒x(x+14)−13(x+14)=0
⇒(x+14)(x−13)=0
Either, x+14=0orx−13=0,
⇒x=−14 or x=13
Since the integers are positive, x can only be 13.
∴x+1=13+1=14
Therefore, two consecutive positive integers will be 13 and 14.