Given the equation:
[tex]\[ x^2 - 3x + 2 = \sqrt{x - 2} + 2 \][/tex]
We want to approximate the solution to this equation using three iterations of successive approximation. We will start with the initial value [tex]\( x_0 = 2.5 \)[/tex] which we obtained from the graph. Here's a detailed step-by-step process:
1. Starting Point:
[tex]\[
x_0 = 2.5
\][/tex]
2. First Iteration:
We need to update the value of [tex]\( x \)[/tex] based on our approximation method. After the first iteration, the new value becomes:
[tex]\[
x_1 \approx -0.3311
\][/tex]
3. Second Iteration:
Using the new value from the first iteration:
[tex]\[
x_2 \approx 0.772 - 0.575i
\][/tex]
4. Third Iteration:
Using the new value from the second iteration:
[tex]\[
x_3 \approx -2.21 + 0.571i
\][/tex]
After three iterations of successive approximation, our approximations are [tex]\[2.5, -0.3311, 0.772 - 0.575i, -2.21 + 0.571i\][/tex]
From the given options, none of them directly map to our detailed steps of approximation. However, understanding the approach, you should have an idea of how methods for numerical solutions aid in converging to a result step-by-step.
