To solve the equation
[tex]\[
x^2 - 3x + 2 = \sqrt{x - 2} + 2
\][/tex]
using successive approximation, we need to choose a starting point based on the given options and iterate to approximate a solution.
Let's proceed with the detailed steps:
1. Select the Initial Value:
Given the multiple choice options, we select the initial value from option A:
[tex]\[
x_0 = \frac{53}{16} \approx 3.3125
\][/tex]
2. Define the Functions:
We start by defining two functions:
[tex]\( f(x) = x^2 - 3x + 2 \)[/tex]
[tex]\( g(x) = \sqrt{x - 2} + 2 \)[/tex]
3. Fixed-Point Iteration:
We iterate to refine our approximation. For each iteration:
[tex]\[
x_{n+1} = g(x_n)
\][/tex]
4. Iterations:
- Iteration 1:
[tex]\[
x_1 = g(x_0) = \sqrt{3.3125 - 2} + 2 = \sqrt{1.3125} + 2 \approx 3.1456439237389597
\][/tex]
- Iteration 2:
[tex]\[
x_2 = g(x_1) = \sqrt{3.1456439237389597 - 2} + 2 = \sqrt{1.1456439237389597} + 2 \approx 3.070347571464036
\][/tex]
- Iteration 3:
[tex]\[
x_3 = g(x_2) = \sqrt{3.070347571464036 - 2} + 2 = \sqrt{1.070347571464036} + 2 \approx 3.0345760346460944
\][/tex]
After three iterations, we can approximate the root of the equation as:
[tex]\[
x \approx 3.0345760346460944
\][/tex]
So the initial value choice [tex]\( x \approx \frac{53}{16} \)[/tex] leads us to a solution. Therefore, the correct starting point based on the given multiple choice options is:
[tex]\[
\boxed{A. \, x \approx \frac{53}{16}}
\][/tex]
