Use the false position method to find the root of 2x² - 3 = 0

The false position method, also known as the regula falsi method, is an iterative method for finding roots of a function. The method requires two initial guesses, \(x_0\) and \(x_1\), such that \(f(x_0)\) and \(f(x_1)\) have opposite signs. We use the formula for the next approximation \(x_2\) given by:

\[ x_2 = x_1 - \frac{f(x_1)(x_1 - x_0)}{f(x_1) - f(x_0)} \]

For the given equation \(2x^2 - 3 = 0\), let \(f(x) = 2x^2 - 3\). We need to find the roots of this function.

Let's follow the steps of the false position method:

1. **Choose initial guesses:** Let's select \(x_0 = 1\) and \(x_1 = 2\).

2. **Evaluate the function at these points:**
   \[
   f(x_0) = 2(1)^2 - 3 = 2 - 3 = -1
   \]
   \[
   f(x_1) = 2(2)^2 - 3 = 8 - 3 = 5
   \]

3. **Calculate \(x_2\) using the false position formula:**
   \[
   x_2 = x_1 - \frac{f(x_1)(x_1 - x_0)}{f(x_1) - f(x_0)}
   \]
   \[
   x_2 = 2 - \frac{5(2 - 1)}{5 - (-1)}
   \]
   \[
   x_2 = 2 - \frac{5 \times 1}{6}
   \]
   \[
   x_2 = 2 - \frac{5}{6}
   \]
   \[
   x_2 = 2 - 0.8333 = 1.1667
   \]

4. **Evaluate the function at the new point \(x_2\):**
   \[
   f(x_2) = 2(1.1667)^2 - 3
   \]
   \[
   f(x_2) = 2(1.3611) - 3
   \]
   \[
   f(x_2) = 2.7222 - 3 = -0.2778
   \]

Since \(f(x_2)\) has opposite signs to \(f(x_1)\), we update our guesses: now \(x_0 = 1.1667\) and \(x_1\) remains 2.

5. **Repeat the process:**

   Calculate the next approximation \(x_3\):
   \[
   x_3 = x_1 - \frac{f(x_1)(x_1 - x_0)}{f(x_1) - f(x_0)}
   \]
   \[
   x_3 = 2 - \frac{5(2 - 1.1667)}{5 - (-0.2778)}
   \]
   \[
   x_3 = 2 - \frac{5 \times 0.8333}{5.2778}
   \]
   \[
   x_3 = 2 - \frac{4.1665}{5.2778}
   \]
   \[
   x_3 = 2 - 0.7896 = 1.2104
   \]

6. **Evaluate the function at the new point \(x_3\):**
   \[
   f(x_3) = 2(1.2104)^2 - 3
   \]
   \[
   f(x_3) = 2(1.4651) - 3
   \]
   \[
   f(x_3) = 2.9302 - 3 = -0.0698
   \]

Continue this iterative process until \(f(x_n)\) is sufficiently close to zero (within a desired tolerance).

**Summary:**
Using the false position method iteratively refines the estimate of the root. After several iterations, the values will converge to the root of the equation \(2x^2 - 3 = 0\). The exact root can be found by continuing this process or switching to another method for further refinement.

Would you like more details or have any questions?

Here are 8 related questions to expand your understanding:

1. What is the main difference between the false position method and the bisection method?
2. How does the choice of initial guesses affect the convergence of the false position method?
3. What are the potential disadvantages of using the false position method?
4. How can you determine if the false position method is converging too slowly?
5. Can the false position method be used for functions with multiple roots? If so, how?
6. How does the false position method compare with Newton's method in terms of speed and reliability?
7. What modifications can be made to the false position method to improve its performance?
8. How can you implement the false position method in a computer program?

**Tip:** When using iterative methods like the false position method, it's important to establish a stopping criterion, such as a maximum number of iterations or a desired tolerance level for the function value. This helps ensure that the method terminates within a reasonable timeframe and provides an accurate approximation of the root.

Learn how to apply the false position method to find the root of the equation 2x² - 3 = 0 step-by-step. This problem involves numerical analysis and root-finding methods, with detailed explanations of the false position method formula and its application. Suitable for advanced high school students.

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