First, we need to expand the expression on the left side of the equation:

(x+7)(x-7) = x^2 - 49

Now we have:

(x^2 - 49) - (x-3)^2 = 53

Expanding (x-3)^2:
(x-3)^2 = x^2 - 6x + 9

Substitute the expanded expressions back into the equation:
(x^2 - 49) - (x^2 - 6x + 9) = 53
x^2 - 49 - x^2 + 6x - 9 = 53
6x - 58 = 53

Now, solve for x:
6x = 111
x = 111/6
x = 18.5

Therefore, the solution to the equation is x = 18.5.