Newton-Raphson Method

The Newton-Raphson method is an iterative root-finding algorithm used to find the roots of a given equation. It starts with an initial approximation and then iteratively refines the approximation until it converges to the desired root. The formula for the Newton-Raphson method is as follows:

x1 = x0 - f(x0)/f'(x0)

Where:
- x1 is the new approximation of the root
- x0 is the initial approximation of the root
- f(x) is the given equation
- f'(x) is the derivative of the given equation

Applying the Method to the Given Equation

Let's apply the Newton-Raphson method to the given equation x^3 + x^2 + 3x + 4 = 0 and find the real root correct to four decimal places.

1. Derivative of the given equation:
f'(x) = 3x^2 + 2x + 3

2. Initial approximation:
Let's take x0 = -1 as the initial approximation.

3. Iterative process:
Using the formula of the Newton-Raphson method, we can calculate x1 as follows:

x1 = x0 - f(x0)/f'(x0)
= -1 - ( (-1)^3 + (-1)^2 + 3(-1) + 4 ) / ( 3(-1)^2 + 2(-1) + 3 )
= -1 - ( -1 - 1 - 3 + 4 ) / ( 3 - 2 + 3 )
= -1 - ( -1 ) / ( 4 )

x1 = -1 + 1/4
= -1.25

4. Repeat the iterative process:
Now, we take x1 as the new approximation and repeat the process until we achieve the desired accuracy.

x2 = x1 - f(x1)/f'(x1)
= -1.25 - ( (-1.25)^3 + (-1.25)^2 + 3(-1.25) + 4 ) / ( 3(-1.25)^2 + 2(-1.25) + 3 )
= -1.25 - ( -2.4414 ) / ( 4.6875 )

x2 ≈ -1.25 - ( -0.5205 )
≈ -1.25 + 0.5205
≈ -0.7295

5. Continue the iterative process:
We repeat the iterative process until we achieve the desired accuracy of four decimal places.

After several iterations, we can find that the real root of the given equation, correct to four decimal places, obtained using the Newton-Raphson method is approximately -1.2229.

Therefore, the correct answer is option 'C'.