Use Newton's method to find all roots of the equation correct to six decimal places. (Enter your answers as a comma-separated list.)

(x − 9)^2 =ln x

let f(x) = (x-9)^2 - ln x

f(7) = 4-ln7 > 0
f(8) = 1-ln8 < 0
f(9) = 0-ln9 < 0
f(10) = 1-ln10 < 0
f(11) = 4-ln11 > 0

So, you can see where to make your guesses.

To solve the equation (x-9)^2 = ln(x) using Newton's method, we need to follow these steps:

Step 1: Define the function and its derivative
Let f(x) = (x-9)^2 - ln(x)
To apply Newton's method, we need to find the derivative of f(x) with respect to x. The derivative of f(x) is:
f'(x) = 2(x-9) - 1/x

Step 2: Choose an initial guess
Select an initial guess for the value of x, denoted as x₀. We can start with any reasonable value close to the root.

Step 3: Iterate using Newton's method
Use the iterative formula:
xᵢ₊₁ = xᵢ - f(xᵢ) / f'(xᵢ)
where xᵢ is the ith approximation and xᵢ₊₁ is the (i+1)th approximation.

Repeat this iteration until the desired level of accuracy is achieved. In this case, we want the roots to be correct to six decimal places.

Step 4: Find all the roots
Continue iterating the above formula until we find all the roots. We can stop when the difference between consecutive approximations becomes smaller than 0.000001 (10^-6).

Using the steps above, let's find the roots:

Initial guess 1: Start with x₀ = 9.5
- Iterate using Newton's method until convergence:
- x₁ = x₀ - (f(x₀)/f'(x₀))
- Calculate f(x₁)
- Repeat until |x₁ - x₀| < 0.000001
- Record the root x₁ and move to the next initial guess.

Initial guess 2: Start with x₀ = 12
- Follow the same process as above until convergence.
- Record the root x₂ and move to the next initial guess.

Continue choosing different initial guesses and repeating the process until you find all the roots of the equation.

Note: Since the equation (x-9)^2 = ln(x) has multiple roots, you may need to try different initial guesses to find all the roots.