Find the terms containing x^2 in the expression (3x-1/2x)^8

To find the term containing \(x^2\) in the expansion of \((3x - \frac{1}{2x})^8\), we'll again 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:
- \(a = 3x\)
- \(b = -\frac{1}{2x}\)
- \(n = 8\)

### Step 2: General Term in the Expansion
The general term in the expansion is given by:
\[
T_{k+1} = \binom{8}{k} (3x)^{8-k} \left(-\frac{1}{2x}\right)^k
\]

### Step 3: Simplify the General Term
Simplifying \(T_{k+1}\):
\[
T_{k+1} = \binom{8}{k} \cdot (3x)^{8-k} \cdot \left(-\frac{1}{2x}\right)^k
\]
\[
T_{k+1} = \binom{8}{k} \cdot 3^{8-k} \cdot x^{8-k} \cdot \frac{(-1)^k}{2^k} \cdot x^{-k}
\]
\[
T_{k+1} = \binom{8}{k} \cdot \frac{3^{8-k} \cdot (-1)^k}{2^k} \cdot x^{8-k-k}
\]
\[
T_{k+1} = \binom{8}{k} \cdot \frac{3^{8-k} \cdot (-1)^k}{2^k} \cdot x^{8-2k}
\]

### Step 4: Find the Value of \(k\) for \(x^2\)
To find the term containing \(x^2\), set the exponent of \(x\) equal to 2:
\[
8 - 2k = 2
\]
Solve for \(k\):
\[
8 - 2k = 2
\]
\[
2k = 6
\]
\[
k = 3
\]

### Step 5: Substitute \(k = 3\) into the General Term
Now substitute \(k = 3\) back into the general term:
\[
T_{4} = \binom{8}{3} \cdot \frac{3^{8-3} \cdot (-1)^3}{2^3} \cdot x^2
\]
\[
T_{4} = \binom{8}{3} \cdot \frac{3^5 \cdot (-1)}{8} \cdot x^2
\]
\[
T_{4} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} \cdot \frac{243 \cdot (-1)}{8} \cdot x^2
\]
\[
T_{4} = 56 \cdot \frac{-243}{8} \cdot x^2
\]
\[
T_{4} = -1701x^2
\]

### Final Answer
The term containing \(x^2\) in the expansion of \((3x - \frac{1}{2x})^8\) is:
\[
-1701x^2
\]

Would you like any more details or have any questions?

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**Related Questions:**

1. How do you find the coefficient of a specific term in a binomial expansion?
2. What is the term containing \(x^4\) in this expansion?
3. How would the expansion change if the exponent were a different value?
4. What are the first and last terms in the expansion?
5. Can you identify a pattern in the coefficients of terms in the binomial expansion?

**Tip:** When dealing with binomial expansions that involve fractions or negative terms, pay close attention to how these factors influence the coefficients and the signs of each term.

Explore how to find the term containing x^2 in the expansion of (3x - 1/2x)^8 using the Binomial Theorem. This detailed solution covers step-by-step calculations and combinatorial analysis. Suitable for high school students studying algebra and binomial expansions.

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