This problem uses an iterative process to approximate a solution to the equation $x^3 - 5x^2 - 11 = 0$ to two decimal places (2 d.p.). The steps involve the following process:

1. Start with an initial value of $x = 5$.
2. Use the formula $x = 5 + \frac{11}{x^2}$.
3. Calculate the new $x$, round it to 2 d.p., and compare it with the previous $x$. If they match, the solution is found. Otherwise, repeat using the updated $x$.

Let me work through this step-by-step to find the approximate solution:

Step 1: Start with $x = 5$.

Step 2: Compute $5 + \frac{11}{x^2}$ with $x = 5$.

$x = 5 + \frac{11}{5^2} = 5 + \frac{11}{25} = 5 + 0.44 = 5.44$

Step 3: Round to 2 d.p. and compare.

• Rounded $x = 5.44$.
• Since this does not match the initial $x = 5.00$, repeat the process using $x = 5.44$.

Next Iteration (using $x = 5.44$):

Step 2: Compute $5 + \frac{11}{x^2}$ with $x = 5.44$.

$x = 5 + \frac{11}{(5.44)^2}$ First, calculate $5.44^2 = 29.5936$, $x = 5 + \frac{11}{29.5936} = 5 + 0.3716 = 5.3716$

Step 3: Round to 2 d.p. and compare.

• Rounded $x = 5.37$.
• Since this does not match the previous $x = 5.44$, repeat using $x = 5.37$.

Next Iteration (using $x = 5.37$):

Step 2: Compute $5 + \frac{11}{x^2}$ with $x = 5.37$.

$x = 5 + \frac{11}{(5.37)^2}$ First, calculate $5.37^2 = 28.8369$, $x = 5 + \frac{11}{28.8369} = 5 + 0.3815 = 5.3815$

Step 3: Round to 2 d.p. and compare.

• Rounded $x = 5.38$.
• Since this does not match the previous $x = 5.37$, repeat using $x = 5.38$.

Next Iteration (using $x = 5.38$):

Step 2: Compute $5 + \frac{11}{x^2}$ with $x = 5.38$.

$x = 5 + \frac{11}{(5.38)^2}$ First, calculate $5.38^2 = 28.9444$, $x = 5 + \frac{11}{28.9444} = 5 + 0.3801 = 5.3801$

Step 3: Round to 2 d.p. and compare.

• Rounded $x = 5.38$.
• Since this matches the previous $x = 5.38$, the process stops.

Final Answer:

The approximate solution to $x^3 - 5x^2 - 11 = 0$ is $x = 5.38$ to 2 d.p.

Would you like me to verify this solution or explain any specific step further?

Here are some related questions to expand on:

1. How does the iterative process converge to a solution?
2. Can this method fail for certain equations or initial values?
3. How can we verify if $x = 5.38$ is indeed a solution to the given equation?
4. What is the geometric interpretation of this iterative method?
5. Are there alternative methods to solve this equation?

Tip: Iterative methods often rely on choosing a good initial value to ensure convergence to the correct solution!