To find the second term of the expansion of \((2x - 3)^6\) using the binomial theorem, we can utilize the formula for the \(k\)-th term in the expansion:
\[ T_k = \binom{n}{k} a^{n-k} b^k \]
where:
- \(n\) is the exponent (in this case, \(n=6\)),
- \(a\) is the first term in the binomial expression (here, \(a=2x\)),
- \(b\) is the second term in the binomial expression (here, \(b=-3\)), and
- \(k\) is the term number we are calculating (recall that \(k\) starts at 0).
The second term corresponds to \(k=1\). Thus, we will calculate \(T_1\):
\[ T_1 = \binom{6}{1} (2x)^{6-1} (-3)^1 \]
Calculating each component:
-
The binomial coefficient: \[ \binom{6}{1} = 6 \]
-
The term \((2x)^{5}\): \[ (2x)^5 = 32x^5 \]
-
The term \((-3)^1\): \[ (-3) = -3 \]
Now, substitute these values back into the term formula:
\[ T_1 = 6 \cdot 32x^5 \cdot (-3) \]
Now perform the multiplication:
\[ T_1 = 6 \cdot 32 \cdot (-3) x^5 \] \[ T_1 = 192 \cdot (-3) x^5 \] \[ T_1 = -576 x^5 \]
Thus, the second term of the expansion is:
\[ \boxed{-576x^5} \]