The equation given is:

$8x^3 + x^2 - 2x + 7 = 0$

We are tasked with finding the solution for $r$ in the interval $-2 \leq r \leq -1$, correct to two decimal places.

Approach:

To approximate the root of the equation, we can use a numerical method such as the bisection method, which is effective for finding roots in a specified interval when the function changes signs.

The bisection method works as follows:

1. Define the function $f(x) = 8x^3 + x^2 - 2x + 7$.
2. Check the signs of $f(-2)$ and $f(-1)$ to ensure there is a root between the two.
3. If the signs are different, split the interval and check again.
4. Repeat until we achieve the desired accuracy.

Let's first check the values of the function at $x = -2$ and $x = -1$.

• $f(-2) = 8(-2)^3 + (-2)^2 - 2(-2) + 7 = -64 + 4 + 4 + 7 = -49$
• $f(-1) = 8(-1)^3 + (-1)^2 - 2(-1) + 7 = -8 + 1 + 2 + 7 = 2$

Since $f(-2) < 0$ and $f(-1) > 0$, there is indeed a root between $-2$ and $-1$.

Now, let's apply the bisection method to narrow down the root. I will calculate the midpoints iteratively until we achieve the required accuracy.

Iterations of the Bisection Method:

1. Start with the interval $[-2, -1]$.
2. Midpoint: $c = \frac{-2 + (-1)}{2} = -1.5$.
   • $f(-1.5) = 8(-1.5)^3 + (-1.5)^2 - 2(-1.5) + 7$
   • Calculate and refine further until the root is approximated to two decimal places.

Let me perform the calculations to find the root. Defining the function for f(x) def f(x): return 8x3 + x2 - 2x + 7

Bisection method implementation to find the root within the interval [-2, -1]

def bisection_method(a, b, tolerance=0.01): # Check if f(a) and f(b) have opposite signs if f(a) * f(b) > 0: return None # No root in the interval

# Start with the bisection process
while (b - a) / 2 > tolerance:
    c = (a + b) / 2  # Midpoint
    if f(c) == 0:  # Found the root
        break
    elif f(a) * f(c) < 0:
        b = c  # Root is in the left half
    else:
        a = c  # Root is in the right half
return (a + b) / 2  # Return the midpoint as the root

Applying bisection method on the given interval [-2, -1]

root = bisection_method(-2, -1, tolerance=0.01) rootThe approximate root of the equation $8x^3 + x^2 - 2x + 7 = 0$ in the interval $-2 \leq r \leq -1$, correct to two decimal places, is:

$r \approx -1.09$

Would you like to explore the solution in more detail or have any further questions on this? Here are some related topics you might be interested in:

1. How to apply other numerical methods like Newton's method for finding roots.
2. How to graph the function to visualize the root.
3. How to determine the convergence of the bisection method.
4. What happens if the function doesn't change signs in the interval.
5. How to solve higher-degree polynomial equations using various methods.

Tip: The bisection method is particularly reliable when you know that the function changes sign within the interval, ensuring that a root exists between the endpoints.