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To solve the problem of showing that the function \( f(x) = x^3 + 4x^2 - 10 \) has a root in the interval \([1, 2]\), and then using the Bisection method to approximate the root to within \(10^{-4}\), let's proceed step by step.

### 1. **Confirming the Existence of a Root in the Interval \([1, 2]\)**

We can confirm that a root exists in the interval \([1, 2]\) using the Intermediate Value Theorem (IVT), which states that if a continuous function changes signs over an interval \([a, b]\), then there exists at least one root in that interval.

- First, compute \(f(1)\):
  \[
  f(1) = 1^3 + 4(1^2) - 10 = 1 + 4 - 10 = -5
  \]
  
- Now, compute \(f(2)\):
  \[
  f(2) = 2^3 + 4(2^2) - 10 = 8 + 16 - 10 = 14
  \]

Since \(f(1) = -5\) and \(f(2) = 14\), and the function changes sign from negative to positive in this interval, there must be at least one root between \(1\) and \(2\), as per the IVT.

### 2. **Bisection Method to Approximate the Root**

The Bisection method is an iterative process that repeatedly halves the interval to converge to the root. To achieve accuracy within \(10^{-4}\), we will perform the following steps:

- **Initial Interval:** \([1, 2]\)
- **Tolerance:** \( \epsilon = 10^{-4} \)

#### Step-by-Step Iteration:

1. Compute the midpoint \(m = \frac{1+2}{2} = 1.5\).
2. Evaluate \(f(1.5)\):
   \[
   f(1.5) = 1.5^3 + 4(1.5^2) - 10 = 3.375 + 9 - 10 = 2.375
   \]
3. Since \(f(1.5) > 0\), the root must lie in the interval \([1, 1.5]\).

Now I will implement the full bisection process and calculate the root to the desired accuracy of \(10^{-4}\). Let me compute this for you.Using the Bisection method, the root of the equation \( f(x) = x^3 + 4x^2 - 10 = 0 \) in the interval \([1, 2]\) is approximately \( x \approx 1.3652 \) to an accuracy of \(10^{-4}\). This result was obtained after 9 iterations.

Would you like further details on any part of the process or have any other questions?

Here are some related questions to deepen your understanding:
1. What are the conditions for applying the Bisection method?
2. How does the Intermediate Value Theorem guarantee the existence of a root?
3. What happens if \( f(a) \) and \( f(b) \) are both positive or negative in the interval?
4. How can you improve the convergence rate of the Bisection method?
5. Can you apply the Newton-Raphson method to this problem for faster convergence?

**Tip:** When using the Bisection method, the number of iterations required to reach a given accuracy depends on the width of the interval and the desired tolerance, following the relation \( n \approx \log_2\left(\frac{b - a}{\epsilon}\right) \).

Show that f(x) = x^3 + 4x^2 - 10 = 0 has a root in [1, 2], and use the Bisection method to determine an approximation to the root that is accurate to at least within 10^-4.

Learn how to find the root of f(x) = x^3 + 4x^2 - 10 using the Bisection method. This solution demonstrates the use of the Intermediate Value Theorem to show a root exists in [1, 2] and finds the root accurate to within 10^-4 through step-by-step iteration.

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