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The problem provided involves applying **Newton's method** to solve the nonlinear equation:

\[
e^{1.4x} = 1.5x^2 - 24
\]

### Recap of Newton's Method:
Newton's method is an iterative process used to approximate roots of a function. For a function \( f(x) \), the iterative formula is:

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

Here, \( f(x) = e^{1.4x} - (1.5x^2 - 24) \), and we need to compute its derivative \( f'(x) \).

### Steps:
1. **Define the function and its derivative:**

   \[
   f(x) = e^{1.4x} - (1.5x^2 - 24)
   \]
   
   The derivative of \( f(x) \) with respect to \( x \) is:

   \[
   f'(x) = 1.4e^{1.4x} - 3.0x
   \]

2. **Given values**: 
   After two iterations, \( x_2 = 4.100137911 \).

3. **Compute \( x_3 \)**:
   Use the Newton's method formula to compute \( x_3 \):

   \[
   x_3 = x_2 - \frac{f(x_2)}{f'(x_2)}
   \]

4. **Estimate the error**: The error is typically calculated as:

   \[
   \text{Error} = |x_3 - x_2|
   \]

You are asked to maintain **at least 10 significant digits** and report the result rounded to 8 decimal places.

Would you like me to walk through the calculations in more detail to find \( x_3 \) and the error? 

Here are 5 related questions that can deepen your understanding:
1. What happens if Newton's method does not converge?
2. How does the choice of initial guess \( x_0 \) affect the convergence of Newton's method?
3. Can Newton's method fail to find a root, and why?
4. What is the significance of the derivative \( f'(x) \) in Newton's method?
5. How can you modify Newton’s method if \( f'(x) \) is zero or very small?

**Tip:** When using Newton's method, always check that the function's derivative is not zero at your approximation, as division by zero would cause an error.

Consider the solution of the following nonlinear equation using Newton’s method: e^(1.4x) = 1.5x^2 - 24. After two iterations, the estimated solution is x2 = 4.100137911. Using Newton's method, find the next estimate of the solution (x3) and use this to determine an absolute estimate of the error in x2. Maintain at least 10 significant digits and round your answers to 8 decimal places.

Learn how to apply Newton’s method to solve the nonlinear equation e^(1.4x) = 1.5x^2 - 24. This detailed walkthrough explains how to estimate the next solution (x3) and calculate the error. With precise significant digits and error estimation, this guide is perfect for undergraduate students.

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