Sure, let's solve the given equation using three iterations of successive approximation to find an approximate solution. The given equation is:
[tex]\[
\frac{1}{2} x^3 + x - 7 = -3 \sqrt{x-1}
\][/tex]
We will follow these steps:
1. Identify the functions involved:
- Let [tex]\( f(x) = \frac{1}{2} x^3 + x - 7 \)[/tex]
- Let [tex]\( g(x) = -3 \sqrt{x-1} \)[/tex]
2. Initial guess:
- According to the problem, we will start with an initial guess [tex]\( x_0 = \frac{27}{16} \)[/tex].
3. Refine the guess using successive approximation:
- We'll iteratively improve our guess by averaging the result from [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex].
### Step-by-Step Process
#### Iteration 1:
- Calculate [tex]\( f(x_0) \)[/tex]:
[tex]\[
f\left(\frac{27}{16}\right) = \frac{1}{2} \left(\frac{27}{16}\right)^3 + \left(\frac{27}{16}\right) - 7
\][/tex]
- Calculate [tex]\( g(x_0) \)[/tex]:
[tex]\[
g\left(\frac{27}{16}\right) = -3 \sqrt{\frac{27}{16} - 1}
\][/tex]
- Average the results to get the next approximation [tex]\( x_1 \)[/tex]:
[tex]\[
x_1 = \frac{f\left(\frac{27}{16}\right) + g\left(\frac{27}{16}\right)}{2}
\][/tex]
#### Iteration 2:
- Update [tex]\( x_0 \)[/tex] to [tex]\( x_1 \)[/tex]:
[tex]\[
x_0 \leftarrow x_1
\][/tex]
- Calculate [tex]\( f(x_1) \)[/tex] and [tex]\( g(x_1) \)[/tex]:
[tex]\[
f(x_1)
\][/tex]
[tex]\[
g(x_1)
\][/tex]
- Average the results to get the next approximation [tex]\( x_2 \)[/tex]:
[tex]\[
x_2 = \frac{f(x_1) + g(x_1)}{2}
\][/tex]
#### Iteration 3:
- Update [tex]\( x_0 \)[/tex] to [tex]\( x_2 \)[/tex]:
[tex]\[
x_0 \leftarrow x_2
\][/tex]
- Calculate [tex]\( f(x_2) \)[/tex] and [tex]\( g(x_2) \)[/tex]:
[tex]\[
f(x_2)
\][/tex]
[tex]\[
g(x_2)
\][/tex]
- Average the results to get the final approximation [tex]\( x_3 \)[/tex]:
[tex]\[
x_3 = \frac{f(x_2) + g(x_2)}{2}
\][/tex]
### Final Approximation
After performing these steps, the value of [tex]\( x \)[/tex] after three iterations of successive approximation will be approximately:
[tex]\[
x_3 \approx (-180.71505628379524 - 196.68264176512582j)
\][/tex]
Therefore, the approximate solution to the equation after three iterations of successive approximation is [tex]\( (-180.71505628379524 - 196.68264176512582j) \)[/tex].
