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

Answer

To solve the problem where \( x^2 + y^2 = 37 \) and \( xy = 6 \), and we need to find \( x - y \), we can follow these steps:

### Step 1: Use the identity for \( (x - y)^2 \)
We know the identity:
\[ (x - y)^2 = x^2 + y^2 - 2xy \]
This identity allows us to express \( x - y \) in terms of \( x^2 + y^2 \) and \( xy \).

### Step 2: Substitute the known values
From the problem, we have:
- \( x^2 + y^2 = 37 \)
- \( xy = 6 \)

Now substitute these values into the identity:
\[ (x - y)^2 = 37 - 2 \cdot 6 \]

### Step 3: Calculate \( (x - y)^2 \)
Now perform the calculation:
\[ (x - y)^2 = 37 - 12 = 25 \]

### Step 4: Take the square root
To find \( x - y \), we take the square root of both sides:
\[ x - y = \sqrt{25} \]
This gives us:
\[ x - y = 5 \quad \text{or} \quad x - y = -5 \]

### Final Answer
Thus, the possible values for \( x - y \) are:
\[ x - y = 5 \quad \text{or} \quad x - y = -5 \]