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To find a negative root of the equation 

\[
e^{0.5x} - x^2 - 3x = 0
\]

using Newton's Method, we can follow these steps:

### Step 1: Set up Newton's Method formula
Newton's Method iteratively refines an approximation for the root of a function \( f(x) = 0 \). The formula for the iteration is:

\[
x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}
\]

where:
- \( f(x) = e^{0.5x} - x^2 - 3x \)
- \( f'(x) \) is the derivative of \( f(x) \).

### Step 2: Compute \( f(x) \) and \( f'(x) \)
1. **Function \( f(x) \):**
   \[
   f(x) = e^{0.5x} - x^2 - 3x
   \]

2. **Derivative \( f'(x) \):**
   \[
   f'(x) = \frac{d}{dx}\left(e^{0.5x} - x^2 - 3x\right) = 0.5 e^{0.5x} - 2x - 3
   \]

### Step 3: Choose an Initial Guess
Since we are looking for a negative root, let’s start with an initial guess \( x_0 \) in the negative range. A reasonable choice might be \( x_0 = -1 \), but we could also try other negative values if this does not converge quickly.

### Step 4: Perform the Iterations
For each iteration:
1. Calculate \( f(x_n) \) and \( f'(x_n) \).
2. Update \( x_n \) using the formula \( x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \).
3. Repeat until \( x_n \) converges to a stable value (typically when \( |x_{n+1} - x_n| \) is below a small tolerance, such as \( 10^{-6} \)).

Let me know if you'd like to see the iterations step-by-step, or if you have any specific values in mind for the initial guess.

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Would you like further assistance with the calculations?

### Related Questions
1. How does Newton's Method behave with different initial guesses?
2. What are the limitations of Newton's Method?
3. Can we apply Newton's Method to a function with no real roots?
4. How does the convergence of Newton's Method change with respect to the derivative's behavior?
5. What are alternative methods to Newton's Method for finding roots?

### Tip:
Always ensure that \( f'(x) \neq 0 \) in the denominator during Newton’s Method iterations to avoid undefined behavior.

Use Newton's method to find a negative root of the equation e^(0.5x) - x^2 - 3x = 0.

Learn how to use Newton's method to find a negative root for the equation e^(0.5x) - x^2 - 3x = 0. This tutorial covers step-by-step instructions including function setup, derivative calculation, and iterative approximation. Suitable for students studying undergraduate calculus.

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