https://file.aiquickdraw.com/mathbot/file/1726480642672xt9d3go7.jpg

The image asks to evaluate \( \sqrt{12} \) to four decimal places using the **Newton-Raphson method**.

### Newton-Raphson Method Overview:
The Newton-Raphson method is an iterative technique used to find successively better approximations of the roots (or zeros) of a real-valued function.

For this problem, we want to solve \( f(x) = x^2 - 12 = 0 \) because the square root of 12 is the root of this equation.

### Formula:
The general formula for Newton-Raphson is:
\[
x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}
\]
Where:
- \( f(x) = x^2 - 12 \)
- \( f'(x) = 2x \)

Now, let's calculate \( \sqrt{12} \) using this method.

1. **Initial guess**: We start with an initial guess, say \( x_0 = 3.5 \), because \( \sqrt{12} \) is between 3 and 4.

I'll compute the first few iterations to approximate \( \sqrt{12} \) to four decimal places.Using the Newton-Raphson method, the approximation for \( \sqrt{12} \) to four decimal places is **3.4641**. This result was obtained after 2 iterations.

Would you like more details on how the iterations were performed or any related information?

Here are five related questions you might find useful:
1. How does the Newton-Raphson method converge so quickly?
2. Can Newton-Raphson be used for complex numbers?
3. What are the limitations of the Newton-Raphson method?
4. How does this method compare to other root-finding algorithms, such as the bisection method?
5. How would you apply this method to functions other than square roots?

**Tip:** Always choose an initial guess close to the actual root to ensure faster convergence when using the Newton-Raphson method.

Evaluate √12 to four decimal places by Newton's Raphson formula.

This solution shows how to evaluate the square root of 12 using the Newton-Raphson method, an iterative technique for approximating roots. The solution reaches 3.4641 after two iterations, with key formulas and steps explained in detail. Suitable for Grades 10-12 and early college-level math.

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