In positive numbers, the solution is x = 12:


    12^2 + (12-7)^2 = 144 + 25 = 169 = 13^2.


The triple (5,12,13) is so called Pythagorean triple - next after the Pythagorean triple (3,4,5).


In negative numbers, the solution is x = -5; then


    (-5)^2 + (-5-7)^2 = 25 + 144 = 169 = 13^2.


You can solve the original equation formally


    x^2 + x^2 - 14x + 49 = 169

    2x^2 - 14x - 120 = 0

     x^2 - 7x  -  60 = 0      (*)


Next, factor the left side


    (x-12)*(x + 5) = 0,


giving you the same two roots  x= 12  and  x= -5.


Or, alternatively, you can apply the quadratic formula to solve equation (*).