Find the coefficient of x^2 in the expansion of (2x - 3)^4.

Find the coefficient of x^2 in the expansion of (2x - 3)^4.

Step 1: Identify the expression we need to expand, which is (2x - 3)^4.
Step 2: Recall the binomial theorem, which states that (a + b)^n = sum of (nCk * a^(n-k) * b^k) for k from 0 to n.
Step 3: In our case, a = 2x, b = -3, and n = 4.
Step 4: We want to find the coefficient of x^2. This means we need to find the term where the power of x is 2.
Step 5: To get x^2 from (2x)^k, we need k = 2, because (2x)^2 = 4x^2.
Step 6: The remaining power of the binomial (2x - 3) is 4 - k = 4 - 2 = 2, so we will use (-3)^2.
Step 7: Calculate the binomial coefficient for k = 2, which is 4C2. This is calculated as 4! / (2!(4-2)!) = 6.
Step 8: Now, calculate the coefficient: 4C2 * (2)^2 * (-3)^2 = 6 * 4 * 9.
Step 9: Perform the multiplication: 6 * 4 = 24, and then 24 * 9 = 216.
Step 10: The coefficient of x^2 in the expansion of (2x - 3)^4 is 216.

Binomial Theorem – The binomial theorem provides a formula for expanding expressions of the form (a + b)^n, where the coefficients can be determined using combinations.
Coefficient Extraction – Identifying the specific coefficient of a term in a polynomial expansion, particularly focusing on the powers of the variable.