The Newton-Raphson method is a widely used iterative method to find the roots of a given equation. It is based on the idea that if we can approximate the root of a given equation, we can use that approximation to find an even better approximation.

The first step in the Newton-Raphson method is to choose an initial guess for the root of the equation. This initial guess can be any value that we think might be close to the actual root. For example, let’s consider the equation x^2 – 6x + 9 = 0. To find the roots of this equation using the Newton-Raphson method, we need to choose an initial guess for the root. Let’s say that we choose x = 3 as our initial guess.

The next step is to use this initial guess to find an improved approximation for the root. We do this by using the equation f(x) = x – f(x)/f'(x). In this equation, f(x) is the function we are trying to find the root of, and f'(x) is the derivative of that function.

For our example equation x^2 – 6x + 9 = 0, the derivative of the function is 2x – 6. So, to find an improved approximation for the root using the Newton-Raphson method, we need to plug in our initial guess of x = 3 and the derivative of the function (2x – 6) into the equation f(x) = x – f(x)/f'(x). This gives us:

x – (x^2 – 6x + 9)/(2x – 6) = 3 – (3^2 – 63 + 9)/(23 – 6) = 3 – (9 – 18 + 9)/(6 – 6) = 3 – 0 = 3

So, our improved approximation for the root is x = 3.

We can continue this process of finding improved approximations for the root until we get a value that is close enough to the actual root. For example, let’s say that we want to find the root of the equation x^3 – x^2 – x – 1 = 0. We can use the Newton-Raphson method to find an improved approximation for the root as follows:

Initial guess: x = 1

Improved approximation: x = 1 – (1^3 – 1^2 – 1 – 1)/(31^2 – 21 – 1) = 1 – (-1)/(1) = 1 – (-1) = 2

Improved approximation: x = 2 – (2^3 – 2^2 – 2 – 1)/(32^2 – 22 – 1) = 2 – (8 – 4 – 2 – 1)/(12 – 4 – 1) = 2 – (1)/(7) = 2 – 0.14 = 1.86

As we can see, each improved approximation is closer to the actual root than the previous one. We can continue this process until we get a value that is close enough to the actual root.

In conclusion, the Newton-Raphson method is a useful iterative method for finding the roots of a given equation. It involves choosing an initial guess for the root and using that guess to find improved approximations for the root. By repeating this process, we can eventually find a value that is close enough to the actual root.