Determine the positive interval using bisection method of x² - x - 6.

The Bisection Method is a root-finding method that repeatedly divides an interval in half and then selects the subinterval in which a root must lie. This process continues until the interval is sufficiently small.

To apply the Bisection Method to the function \( f(x) = x^2 - x - 6 \), we need to find an interval \([a, b]\) where the function changes sign, i.e., \( f(a) \cdot f(b) < 0 \).

1. **Define the function:**
   \[
   f(x) = x^2 - x - 6
   \]

2. **Choose initial interval \([a, b]\):**
   We need to find \(a\) and \(b\) such that \( f(a) \) and \( f(b) \) have opposite signs.

   Let's test the interval \([2, 3]\):
   \[
   f(2) = 2^2 - 2 - 6 = 4 - 2 - 6 = -4
   \]
   \[
   f(3) = 3^2 - 3 - 6 = 9 - 3 - 6 = 0
   \]

   Since \( f(2) \cdot f(3) < 0 \), the interval \([2, 3]\) contains a root.

3. **Apply the Bisection Method:**

   1. Compute the midpoint:
      \[
      c = \frac{a + b}{2} = \frac{2 + 3}{2} = 2.5
      \]
   
   2. Evaluate the function at the midpoint:
      \[
      f(2.5) = 2.5^2 - 2.5 - 6 = 6.25 - 2.5 - 6 = -2.25
      \]
   
   3. Determine the new interval:
      Since \( f(2.5) \) has the same sign as \( f(2) \), we update the interval to \([2.5, 3]\).

4. **Repeat the process:**

   1. Compute the new midpoint:
      \[
      c = \frac{2.5 + 3}{2} = 2.75
      \]
   
   2. Evaluate the function at the new midpoint:
      \[
      f(2.75) = 2.75^2 - 2.75 - 6 = 7.5625 - 2.75 - 6 = -1.1875
      \]
   
   3. Determine the new interval:
      Since \( f(2.75) \) has the same sign as \( f(2.5) \), we update the interval to \([2.75, 3]\).

5. **Continue the process:**

   1. Compute the new midpoint:
      \[
      c = \frac{2.75 + 3}{2} = 2.875
      \]
   
   2. Evaluate the function at the new midpoint:
      \[
      f(2.875) = 2.875^2 - 2.875 - 6 = 8.265625 - 2.875 - 6 = -0.609375
      \]
   
   3. Determine the new interval:
      Since \( f(2.875) \) has the same sign as \( f(2.75) \), we update the interval to \([2.875, 3]\).

6. **Repeat until the interval is sufficiently small:**

   1. Compute the new midpoint:
      \[
      c = \frac{2.875 + 3}{2} = 2.9375
      \]
   
   2. Evaluate the function at the new midpoint:
      \[
      f(2.9375) = 2.9375^2 - 2.9375 - 6 = 8.62890625 - 2.9375 - 6 = -0.30859375
      \]
   
   3. Determine the new interval:
      Since \( f(2.9375) \) has the same sign as \( f(2.875) \), we update the interval to \([2.9375, 3]\).

   4. Continue iterating until the interval length is less than a specified tolerance (e.g., 0.001).

Following this process, we continue iterating, gradually narrowing down the interval where the root lies.

Do you want details on further iterations, or do you have any specific questions?

### Related Questions:
1. What is the main idea behind the Bisection Method?
2. How do you determine the initial interval for the Bisection Method?
3. What are the advantages of the Bisection Method?
4. What are the limitations of the Bisection Method?
5. How does the Bisection Method compare to Newton's Method?
6. Can the Bisection Method be used for functions with multiple roots?
7. How does the convergence rate of the Bisection Method compare to other root-finding methods?
8. What are some practical applications of the Bisection Method in science and engineering?

### Tip:
To ensure accurate results, choose an initial interval where the function changes sign, and continue iterating until the interval is sufficiently small.

Learn how to use the Bisection Method to find the positive interval for the function x² - x - 6. This method involves iteratively narrowing down the interval where the function changes sign until the interval is sufficiently small. Suitable for advanced high school students and college-level mathematics enthusiasts.

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