Let's expand the expression [tex]\(\frac{1}{2}\left[(3a - 3b)^2 + (4c - 2a)^2\right]\)[/tex] step-by-step.
1. First, we expand the square terms inside the parentheses:
[tex]\[ (3a - 3b)^2 \][/tex]
[tex]\[ = (3a - 3b)(3a - 3b) \][/tex]
[tex]\[ = 3a \cdot 3a - 3a \cdot 3b - 3b \cdot 3a + 3b \cdot 3b \][/tex]
[tex]\[ = 9a^2 - 9ab - 9ab + 9b^2 \][/tex]
[tex]\[ = 9a^2 - 18ab + 9b^2 \][/tex]
Similarly, for the second square term:
[tex]\[ (4c - 2a)^2 \][/tex]
[tex]\[ = (4c - 2a)(4c - 2a) \][/tex]
[tex]\[ = 4c \cdot 4c - 4c \cdot 2a - 2a \cdot 4c + 2a \cdot 2a \][/tex]
[tex]\[ = 16c^2 - 8ac - 8ac + 4a^2 \][/tex]
[tex]\[ = 16c^2 - 16ac + 4a^2 \][/tex]
2. Substitute the expanded forms back into the original expression:
[tex]\[ \frac{1}{2}\left[ (9a^2 - 18ab + 9b^2) + (16c^2 - 16ac + 4a^2) \right] \][/tex]
3. Combine like terms inside the parentheses:
[tex]\[ \frac{1}{2}\left[ 9a^2 - 18ab + 9b^2 + 16c^2 - 16ac + 4a^2 \right] \][/tex]
[tex]\[ = \frac{1}{2}\left[ 13a^2 - 18ab - 16ac + 9b^2 + 16c^2 \right] \][/tex]
4. Now, distribute the [tex]\(\frac{1}{2}\)[/tex] across all the terms:
[tex]\[ \frac{1}{2} \cdot 13a^2 - \frac{1}{2} \cdot 18ab - \frac{1}{2} \cdot 16ac + \frac{1}{2} \cdot 9b^2 + \frac{1}{2} \cdot 16c^2 \][/tex]
5. Simplify each term:
[tex]\[ = \frac{13a^2}{2} - 9ab - 8ac + \frac{9b^2}{2} + 8c^2 \][/tex]
So, the expanded form of the given expression [tex]\(\frac{1}{2}\left[(3a - 3b)^2 + (4c - 2a)^2\right]\)[/tex] is:
[tex]\[ \frac{13a^2}{2} - 9ab - 8ac + \frac{9b^2}{2} + 8c^2 \][/tex]
This is the final expanded form.
