What is the term containing x^2 in the expansion of (3x - 4)^6?

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Question: What is the term containing x^2 in the expansion of (3x - 4)^6?

Options:

Correct Answer: -1440

Solution:

The term containing x^2 is given by C(6,2) * (3x)^2 * (-4)^(6-2) = 15 * 9 * 256 = -1440.

What is the term containing x^2 in the expansion of (3x - 4)^6?

What is the term containing x^2 in the expansion of (3x - 4)^6?

Step 1: Identify the expression to expand, which is (3x - 4)^6.
Step 2: Use the Binomial Theorem, which states that (a + b)^n = Σ [C(n, k) * a^(n-k) * b^k] for k = 0 to n.
Step 3: In our case, a = 3x, b = -4, and n = 6.
Step 4: We want the term that contains x^2. This occurs when we choose (3x) two times (k = 2).
Step 5: Calculate the binomial coefficient C(6, 2), which is the number of ways to choose 2 items from 6. C(6, 2) = 6! / (2!(6-2)!) = 15.
Step 6: Calculate (3x)^2. This is (3^2)(x^2) = 9x^2.
Step 7: Calculate (-4)^(6-2), which is (-4)^4. This equals 256.
Step 8: Combine all parts to find the term: C(6, 2) * (3x)^2 * (-4)^(6-2) = 15 * 9 * 256.
Step 9: Calculate 15 * 9 = 135, then multiply by 256 to get 34560.
Step 10: Since we have (-4) raised to an even power, the term is positive, so the final term is 34560x^2.

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