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In the expansion of (2x - 3)^5, what is the coefficient of x^2?

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Question: In the expansion of (2x - 3)^5, what is the coefficient of x^2?

Options:

  1. -90
  2. -120
  3. -150
  4. -180

Correct Answer: -120

Solution: 

The coefficient of x^2 is C(5,2) * (2)^2 * (-3)^3 = 10 * 4 * (-27) = -1080.

In the expansion of (2x - 3)^5, what is the coefficient of x^2?

Practice Questions

Q1
In the expansion of (2x - 3)^5, what is the coefficient of x^2?
  1. -90
  2. -120
  3. -150
  4. -180

Questions & Step-by-Step Solutions

In the expansion of (2x - 3)^5, what is the coefficient of x^2?
Correct Answer: -1080
  • Step 1: Identify the expression to expand, which is (2x - 3)^5.
  • Step 2: Recognize that we need to find the coefficient of x^2 in the expansion.
  • Step 3: 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 4: In our case, a = 2x, b = -3, and n = 5.
  • Step 5: We want the term where x has the power of 2, which means we need to find the term where (2x) is raised to the power of 2.
  • Step 6: This corresponds to k = 3 because n - k = 2 (5 - k = 2). So, k = 3.
  • Step 7: Calculate C(5, 3), which is the number of ways to choose 3 items from 5. C(5, 3) = 5! / (3! * (5-3)!) = 10.
  • Step 8: Calculate (2)^2, which is 4.
  • Step 9: Calculate (-3)^3, which is -27.
  • Step 10: Multiply these values together: Coefficient = C(5, 3) * (2)^2 * (-3)^3 = 10 * 4 * (-27).
  • Step 11: Perform the multiplication: 10 * 4 = 40, and then 40 * (-27) = -1080.
  • Step 12: Conclude that the coefficient of x^2 in the expansion of (2x - 3)^5 is -1080.
  • Binomial Expansion – The process of expanding expressions of the form (a + b)^n using the Binomial Theorem.
  • Coefficients in Binomial Expansion – Understanding how to calculate the coefficients of specific terms in the expansion using combinations.
  • Negative Exponents – Handling negative numbers in the expansion correctly, especially when raised to a power.
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