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.