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

Answer

To solve the problem, we need to find the value of \( a^2 - b^2 \) given that \( a^2 + b^2 = 13 \) and \( ab = 6 \).

### Step-by-Step Solution:

1. **Use the identity for \( a^2 - b^2 \)**:
\[ a^2 - b^2 = (a - b)(a + b) \]
We need to find \( a - b \) and \( a + b \).

2. **Find \( a - b \)**:
We can use the identity:
\[ (a - b)^2 = a^2 + b^2 - 2ab \]
Substitute the known values:
\[ (a - b)^2 = 13 - 2 \times 6 \]
\[ (a - b)^2 = 13 - 12 = 1 \]
Taking the square root:
\[ a - b = \pm 1 \]

3. **Find \( a + b \)**:
We can use the identity:
\[ (a + b)^2 = a^2 + b^2 + 2ab \]
Substitute the known values:
\[ (a + b)^2 = 13 + 2 \times 6 \]
\[ (a + b)^2 = 13 + 12 = 25 \]
Taking the square root:
\[ a + b = \pm 5 \]

4. **Combine the results**:
Now we have two cases for \( a - b \) and \( a + b \):
- Case 1: \( a - b = 1 \) and \( a + b = 5 \)
- Case 2: \( a - b = -1 \) and \( a + b = -5 \)

5. **Calculate \( a^2 - b^2 \)**:
Using \( a^2 - b^2 = (a - b)(a + b) \):
- For Case 1:
\[ a^2 - b^2 = 1 \times 5 = 5 \]
- For Case 2:
\[ a^2 - b^2 = -1 \times -5 = 5 \]

Thus, in both cases, we find that:
\[ \boxed{5} \]