We are asked to use Newton's Method to approximate the zero(s) of the function $f(x) = x - 4\sqrt{x + 3}$ to three decimal places.

Steps for Newton's Method:

1. Initial Guess: Choose a starting value $x_0$ close to where the root might be.
2. Iteration Formula: Newton’s method uses the iterative formula: $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$ where $f'(x)$ is the derivative of $f(x)$.

Step 1: Function and Derivative

The given function is $f(x) = x - 4\sqrt{x + 3}$.

Derivative $f'(x)$:

$f'(x) = 1 - \frac{4}{2\sqrt{x+3}} = 1 - \frac{2}{\sqrt{x+3}}.$

Step 2: Iteration

Using the iterative formula for Newton's method: $x_{n+1} = x_n - \frac{x_n - 4\sqrt{x_n + 3}}{1 - \frac{2}{\sqrt{x_n + 3}}}.$

Step 3: Choose Initial Guess

Let's try an initial guess of $x_0 = 1$, and apply the iterative process until the value stabilizes to three decimal places.

Let me compute the iterations.It seems that the derivative function $f'(x)$ caused a division by zero error at some point, likely because $\sqrt{x+3}$ became too small. This might happen if the guess is too far from the root or if the value of $x+3$ approaches zero.

To avoid this, let's try a larger initial guess for $x_0$, since the term $\sqrt{x + 3}$ becomes more stable for larger values of $x$. I'll try with $x_0 = 2$ and run the calculation again.Using Newton's Method with an initial guess of $x_0 = 2$, the root of the function $f(x) = x - 4\sqrt{x + 3}$ approximates to $x \approx 18.583$, accurate to three decimal places.

Would you like further clarification on the steps, or any additional details?

Here are some follow-up questions for deeper understanding:

1. What are the conditions under which Newton's Method may fail to converge?
2. How would you choose a better initial guess for Newton's Method?
3. How does the rate of convergence of Newton's Method compare to other root-finding methods?
4. Can you apply Newton’s Method to a different function and find its root?
5. What is the significance of the derivative in Newton's Method, and what happens if it is zero?

Tip: Newton's Method requires that the derivative $f'(x)$ not be zero at the starting guess. Always check for this condition before applying the formula.