Which method is commonly used to find the roots of equations?

The Newton-Raphson method is a widely used iterative technique for finding the roots of equations. It is based on the idea of approximating the function with its tangent line and using the point where this tangent line crosses the x-axis as the next approximation of the root. This approach exploits the derivative of the function to rapidly converge to a root.

The method begins with an initial guess for the root and iteratively improves this guess using the formula:

x_{n+1} = x_n - f(x_n)/f'(x_n)

where f is the function for which the root is being sought, and f' is its derivative. This process continues until the difference between successive approximations is within a predefined tolerance level. Its efficiency makes the Newton-Raphson method particularly effective for functions that are well-behaved and where the derivative does not approach zero near the root.

Unlike methods designed for numerical integration, such as the Trapezoidal method and Simpson's rule, the Newton-Raphson method specifically targets root-finding in equations. The Bisection method, another root-finding approach, relies on interval halving and does not utilize derivatives, which can make it slower compared to Newton-Raphson for functions where the derivative