If ` x^(2) + y^(2) = 37 and xy = 6, ` find
` x^(2) - y^(2)`

Answer

To solve the problem, we need to find \( x^2 - y^2 \) given the equations \( x^2 + y^2 = 37 \) and \( xy = 6 \).

### Step-by-Step Solution:

1. **Use the identity for \( x^2 - y^2 \)**:
We know that:
\[ x^2 - y^2 = (x + y)(x - y) \]
Thus, we need to find \( x + y \) and \( x - y \).

2. **Find \( x + y \)**:
We can use the identity:
\[ (x + y)^2 = x^2 + y^2 + 2xy \]
Substituting the known values:
\[ (x + y)^2 = 37 + 2 \cdot 6 \]
\[ (x + y)^2 = 37 + 12 = 49 \]
Taking the square root:
\[ x + y = \pm 7 \]

3. **Find \( x - y \)**:
We can use the identity:
\[ (x - y)^2 = x^2 + y^2 - 2xy \]
Substituting the known values:
\[ (x - y)^2 = 37 - 2 \cdot 6 \]
\[ (x - y)^2 = 37 - 12 = 25 \]
Taking the square root:
\[ x - y = \pm 5 \]

4. **Calculate \( x^2 - y^2 \)**:
Now we can substitute \( x + y \) and \( x - y \) into the formula:
\[ x^2 - y^2 = (x + y)(x - y) \]
Using the values we found:
\[ x^2 - y^2 = (\pm 7)(\pm 5) \]
This gives us:
\[ x^2 - y^2 = 35 \quad \text{or} \quad x^2 - y^2 = -35 \]

Thus, the final answer is:
\[ x^2 - y^2 = \pm 35 \]