The fixed-point iteration method is used to solve equations of the form $x = g(x)$. To use this method for solving $x^2 - 2x - 8 = 0$, we first need to rewrite the equation in the form $x = g(x)$.

Starting from: $x^2 - 2x - 8 = 0$ We can rearrange it to: $x^2 - 2x = 8$ $x^2 = 2x + 8$ $x = \sqrt{2x + 8}$

Thus, we define: $g(x) = \sqrt{2x + 8}$

Given the starting value $x_0 = 5$, we will use the fixed-point iteration method: $x_{n+1} = g(x_n) = \sqrt{2x_n + 8}$

We iterate this until the difference between successive values is less than a specified tolerance, say $10^{-5}$.

Iteration Steps:

Iteration 1:

$x_1 = \sqrt{2x_0 + 8} = \sqrt{2 \cdot 5 + 8} = \sqrt{18} \approx 4.2426$

Iteration 2:

$x_2 = \sqrt{2x_1 + 8} = \sqrt{2 \cdot 4.2426 + 8} = \sqrt{16.4852} \approx 4.0582$

Iteration 3:

$x_3 = \sqrt{2x_2 + 8} = \sqrt{2 \cdot 4.0582 + 8} = \sqrt{16.1164} \approx 4.0139$

Iteration 4:

$x_4 = \sqrt{2x_3 + 8} = \sqrt{2 \cdot 4.0139 + 8} = \sqrt{16.0278} \approx 4.002$

Iteration 5:

$x_5 = \sqrt{2x_4 + 8} = \sqrt{2 \cdot 4.002 + 8} = \sqrt{16.004} \approx 4.0005$

Iteration 6:

$x_6 = \sqrt{2x_5 + 8} = \sqrt{2 \cdot 4.0005 + 8} = \sqrt{16.001} \approx 4.0001$

Iteration 7:

$x_7 = \sqrt{2x_6 + 8} = \sqrt{2 \cdot 4.0001 + 8} = \sqrt{16.0002} \approx 4.000$

Since the difference between $x_7$ and $x_6$ is less than $10^{-5}$, we can stop the iteration process.

Thus, the root of the equation $x^2 - 2x - 8 = 0$ using the fixed-point iteration method, starting from $x_0 = 5$, is approximately $x \approx 4.0000$.

Would you like more details or have any questions?

Here are some related questions for further exploration:

1. How do you choose a suitable function $g(x)$ for fixed-point iteration?
2. What are the conditions for the convergence of the fixed-point iteration method?
3. How does the rate of convergence of fixed-point iteration compare with other iterative methods?
4. What are some common pitfalls to avoid when using fixed-point iteration?
5. Can fixed-point iteration be used for solving systems of nonlinear equations?
6. How does the choice of initial guess affect the convergence of fixed-point iteration?
7. What are the differences between fixed-point iteration and the Newton-Raphson method?
8. How can fixed-point iteration be modified to improve its convergence properties?

Tip: Always verify the convergence criteria and ensure the function $g(x)$ is chosen such that the iterations remain bounded and converge to the desired root.