To find the middle term of the expansion \((2x - \frac{x^2}{4})^9\), we'll use the Binomial Theorem.

### Step 1: Binomial Theorem Formula
The Binomial Theorem states that:
\[
(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
\]
In this case:
- \(a = 2x\)
- \(b = -\frac{x^2}{4}\)
- \(n = 9\)

### Step 2: Middle Term
Since \(n = 9\) (which is odd), the middle term corresponds to the term where \(k = \frac{n+1}{2} = \frac{10}{2} = 5\).

### Step 3: Substitute Values into the Formula
Substituting \(k = 5\) into the binomial expansion formula:
\[
\text{Middle Term} = \binom{9}{5} (2x)^{9-5} \left(-\frac{x^2}{4}\right)^5
\]

### Step 4: Simplify the Expression
Now, let's simplify each part:

1. **Binomial Coefficient:**
   \[
   \binom{9}{5} = \frac{9!}{5!(9-5)!} = \frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2 \times 1} = 126
   \]

2. **Simplifying \( (2x)^4 \):**
   \[
   (2x)^4 = 2^4 \times x^4 = 16x^4
   \]

3. **Simplifying \( \left(-\frac{x^2}{4}\right)^5 \):**
   \[
   \left(-\frac{x^2}{4}\right)^5 = (-1)^5 \times \frac{x^{10}}{4^5} = -\frac{x^{10}}{1024}
   \]

4. **Multiply the components together:**
   \[
   \text{Middle Term} = 126 \times 16x^4 \times \left(-\frac{x^{10}}{1024}\right)
   \]
   Simplifying further:
   \[
   \text{Middle Term} = 126 \times \frac{16}{1024} \times (-x^{14})
   \]
   \[
   \text{Middle Term} = 126 \times \frac{1}{64} \times (-x^{14})
   \]
   \[
   \text{Middle Term} = -\frac{126}{64} \times x^{14} = -\frac{63}{32}x^{14}
   \]

### Final Answer
The middle term of the expansion \((2x - \frac{x^2}{4})^9\) is:
\[
-\frac{63}{32}x^{14}
\]

Would you like any more details or have any questions?

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**Related Questions:**

1. How can you find other specific terms in the binomial expansion?
2. What is the general form of a term in a binomial expansion?
3. How would the process change if the exponent \( n \) was even?
4. What are the coefficients of the first and last terms in this expansion?
5. How do negative exponents affect the terms in a binomial expansion?

**Tip:** The binomial theorem is not only useful for expansions but also for solving combinatorial problems and calculating probabilities in binomial distributions.

Learn how to find the middle term of the binomial expansion (2x - x^2/4)^9 step-by-step. Utilize the Binomial Theorem and Binomial Coefficient to solve the problem. Suitable for high school students studying advanced algebra.

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