There are several ways to approximate the roots of given functions.  The Newton-Raphson Method (an approximation method I was first introduced to in my first year of university) is based on the idea of using the tangent line of a function at a given point to get closer and closer to the unknown root.  There are mathematical cases where there are no exact formulas for solving a homogeneous equation.  Utilization of the Newton-Raphson Method is evidence that calculus-based numerical methods are vital in approximating certain functions.

            The Newton-Raphson Method relies on the tangent to approximate of the function near the point P(x_n, y_n) (assuming y_n = f(x_n) is small) and we let x_n+1 be the value of x where the tangent line at that point will cross the x-intercept (assuming that the tangent line is not zero).  The equation of the tangent line would be:

If we let y = 0,  y_n = f(x_n), we get:

Solving for x, we obtain: