If `a^(2)+b^(2)=40` and `ab=12`, find `a+b and a-b`.

Answer

To solve the problem, we need to find the values of \( a + b \) and \( a - b \) given the equations \( a^2 + b^2 = 40 \) and \( ab = 12 \).

### Step-by-Step Solution:

1. **Use the identity for \( (a + b)^2 \)**:
\[ (a + b)^2 = a^2 + b^2 + 2ab \]
We know \( a^2 + b^2 = 40 \) and \( ab = 12 \).

2. **Substitute the known values into the identity**:
\[ (a + b)^2 = 40 + 2 \times 12 \]
\[ (a + b)^2 = 40 + 24 \]
\[ (a + b)^2 = 64 \]

3. **Take the square root to find \( a + b \)**:
\[ a + b = \sqrt{64} \quad \text{or} \quad a + b = -\sqrt{64} \]
\[ a + b = 8 \quad \text{or} \quad a + b = -8 \]

4. **Use the identity for \( (a - b)^2 \)**:
\[ (a - b)^2 = a^2 + b^2 - 2ab \]

5. **Substitute the known values into the identity**:
\[ (a - b)^2 = 40 - 2 \times 12 \]
\[ (a - b)^2 = 40 - 24 \]
\[ (a - b)^2 = 16 \]

6. **Take the square root to find \( a - b \)**:
\[ a - b = \sqrt{16} \quad \text{or} \quad a - b = -\sqrt{16} \]
\[ a - b = 4 \quad \text{or} \quad a - b = -4 \]

### Final Results:
- \( a + b = 8 \) or \( a + b = -8 \)
- \( a - b = 4 \) or \( a - b = -4 \)