More commonly known as the Newton-Raphson method (at least in the multivariate case, where it also has rapid convergence in many cases), this method follows the tangent to the curve of the function to the X axis, and uses that as the next point.
See the Newton-Raphson method for more details.
It is inaccurate to suggest (but some people do just that) that we need iteration methods only when we have no exact algebraic formula. Especially since your computer uses the Babylonian square root algorithm, a special case of this method, to calculate the function sqrt.