Two consecutive numbers whose square differ by 25

Represent these numbers with x and y.  Their squares are x^2 and y^2, respectively.  The larger square is 25 more than the smaller square:

y^2 = x^2 + 25, assuming that y>x.  If x and y are consecutive, y exceeds x by 1:     x+1=y.

Square x+1 = y, obtaining x^2+2x+1.  Substitute your result (that is, your expression for y^2 in terms of x into the other equation:        
 
 x^2+2x+1 = x^2+25.  Then 2x+1=25; 2x=24; x=12, and y=13.

Check:  Is 13^2  25 units greater than 12^2?  Is 169 25 units greater than 144?  Yes.   So, the two consec. integers are 12 and 13.