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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.

Use Newton's Method to approximate the zero(s) of the function f(x) = x - 4√(x + 3) accurate to three decimal places.

Use Newton's Method to approximate the zero(s) of the function f(x) = x - 4√(x + 3) accurate to three decimal places. This problem involves root approximation techniques and derivative-based iteration methods. Suitable for students studying calculus or numerical methods.

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