X^2 - 2X - 8 = 0
a = 1 b = -2 c= -8
put in the form (x+b/2a)^{2 }and keep the balance
(x-2/2)^{2}-1 - 8 =0
(x - 1)^{2}=9
x- 1 = 3 or x -1 = -3
x = 3 + 1 or x = -3 + 1
x = 4, x = -2
previous solution is correct hopefully this explained where the added 1 came from and how to determine what constant to use to keep the balance of the equation.
x^2-14x-3=0
a = 1 b = -14
(x-14/2)^{2 }=x^{2}-14x +49
since you added 49 on the left then add to the right to keep the balance
(x - 7)² =0 + 49 + 3 = 52
x - 7 = √52
x = 7 + √52 or 7 - √52