Use simple fixed point iteration to locate the root of f(x)=-9x^2+1.7x+2.5 with initial value of x=5

To solve the equation \( f(x) = -9x^2 + 1.7x + 2.5 = 0 \) using simple fixed-point iteration, we need to rewrite the equation in the form \( x = g(x) \). Let's isolate \( x \) to set up the iteration formula.

We can rearrange the equation as follows:

\[
x = g(x) = \frac{1.7x + 2.5}{9x}
\]

For simplicity, we'll now start the fixed-point iteration method:

### Steps for Fixed-Point Iteration:
1. Choose an initial guess for \( x_0 \). Given \( x_0 = 5 \).
2. Use the function \( g(x) \) to calculate the next \( x \) value:
   \[
   x_{n+1} = g(x_n)
   \]
3. Repeat the process until the values converge.

Let's compute the first few iterations manually:

#### Iteration 1:
Using \( x_0 = 5 \):
\[
x_1 = g(x_0) = \frac{1.7(5) + 2.5}{9(5)} = \frac{8.5 + 2.5}{45} = \frac{11}{45} \approx 0.2444
\]

#### Iteration 2:
Now, using \( x_1 = 0.2444 \):
\[
x_2 = g(x_1) = \frac{1.7(0.2444) + 2.5}{9(0.2444)} = \frac{0.41548 + 2.5}{2.1996} \approx 1.3242
\]

#### Iteration 3:
Using \( x_2 = 1.3242 \):
\[
x_3 = g(x_2) = \frac{1.7(1.3242) + 2.5}{9(1.3242)} = \frac{2.25114 + 2.5}{11.918} \approx 0.3779
\]

#### Iteration 4:
Using \( x_3 = 0.3779 \):
\[
x_4 = g(x_3) = \frac{1.7(0.3779) + 2.5}{9(0.3779)} = \frac{0.64243 + 2.5}{3.4011} \approx 0.921
\]

The process will continue until convergence (when \( |x_{n+1} - x_n| \) is sufficiently small).

Would you like me to continue more iterations, or would you like a detailed explanation of each step?

Explore the use of simple fixed point iteration to locate the root of the quadratic equation f(x) = -9x^2 + 1.7x + 2.5. Detailed step-by-step iterations are provided, starting from an initial guess of x = 5. Learn key concepts in fixed point iteration and root-finding methods, suitable for undergraduate-level students.

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