notice in the equation x:=(x+N/x)/2 you yse the variable "x" when the whole purpose of the equation is to get the value of "x". What is the logic in that?
It's an iterated algorithm. It isn't really
x = \frac{x + \frac{N}{x}}{2},
but
x_{n+1} = \frac{x_n + \frac{N}{x_n}}{2}
where N is the number you want the root of. Say, for example, I want \sqrt{2}, so N=2. Then I make the initial guess x_1 = 1, so
x_2 = \frac{x_1 + \frac{2}{x_1}}{2} = \frac{1+\frac{2}{1}}{2} = \frac{3}{2} = 1.5
x_3 = \frac{x_2 + \frac{2}{x_2}}{2} = \frac{\frac{3}{2} + \frac{2}{\frac{3}{2}}}{2} = \frac{17}{12} \approx 1.4167
x_4 = \frac{x_3 + \frac{2}{x_3}}{2} = \frac{\frac{17}{12} + \frac{2}{\frac{17}{12}}}{2} \approx 1.414216
and you can repeat as necessary to get arbitrarily close. The exact root is 1.414213562....
In general, if you want to solve for a root of a differentiable real-valued function f of a single variable x, the algorithm is
x_{n+1} = x_n - \frac{f(x_n)}{f^\prime(x_n)},
though this algorithm is by no means guaranteed to converge (it
is quite easy to put conditions on its convergence though).
Anyways, what exactly do you mean by
...anything can be expressed in a single equation.
?
