If ` a^(2) +b^(2) = 13` and ab= 6 find:
` 3(a+b) ^(2) - 2(a-b)^(2)`

Answer

To solve the problem step by step, we start with the given equations:

1. **Given:**
\[ a^2 + b^2 = 13 \]
\[ ab = 6 \]

2. **We need to find:**
\[ 3(a+b)^2 - 2(a-b)^2 \]

3. **Using the identities:**
\[ (a+b)^2 = a^2 + b^2 + 2ab \]
\[ (a-b)^2 = a^2 + b^2 - 2ab \]

4. **Calculate \((a+b)^2\):**
\[ (a+b)^2 = a^2 + b^2 + 2ab = 13 + 2 \times 6 \]
\[ = 13 + 12 = 25 \]

5. **Calculate \((a-b)^2\):**
\[ (a-b)^2 = a^2 + b^2 - 2ab = 13 - 2 \times 6 \]
\[ = 13 - 12 = 1 \]

6. **Substituting back into the expression:**
\[ 3(a+b)^2 - 2(a-b)^2 = 3 \times 25 - 2 \times 1 \]
\[ = 75 - 2 = 73 \]

7. **Final Answer:**
\[ 73 \]