The number x is 2 more than the number y. If the sum of the squares of x and y is 34, find the product of x and y.
Given : ` x -y = 2 and x^2 + y^2 = 34`
To find the value of xy.

Answer

To solve the problem step by step, we will use the given equations:

1. **Given Equations**:
- \( x - y = 2 \) (Equation 1)
- \( x^2 + y^2 = 34 \) (Equation 2)

2. **Express x in terms of y**:
From Equation 1, we can express \( x \) in terms of \( y \):
\[ x = y + 2 \]

3. **Substitute x in Equation 2**:
Now, substitute \( x \) in Equation 2:
\[ (y + 2)^2 + y^2 = 34 \]

4. **Expand the equation**:
Expanding \( (y + 2)^2 \):
\[ y^2 + 4y + 4 + y^2 = 34 \]
Combine like terms:
\[ 2y^2 + 4y + 4 = 34 \]

5. **Rearrange the equation**:
Move 34 to the left side:
\[ 2y^2 + 4y + 4 - 34 = 0 \]
Simplifying gives:
\[ 2y^2 + 4y - 30 = 0 \]

6. **Divide the entire equation by 2**:
To simplify, divide the entire equation by 2:
\[ y^2 + 2y - 15 = 0 \]

7. **Factor the quadratic equation**:
Now, we will factor the quadratic:
\[ (y + 5)(y - 3) = 0 \]

8. **Find the values of y**:
Setting each factor to zero gives:
\[ y + 5 = 0 \quad \Rightarrow \quad y = -5 \]
\[ y - 3 = 0 \quad \Rightarrow \quad y = 3 \]

9. **Find the corresponding values of x**:
Using \( x = y + 2 \):
- If \( y = -5 \):
\[ x = -5 + 2 = -3 \]
- If \( y = 3 \):
\[ x = 3 + 2 = 5 \]

10. **Calculate the product \( xy \)**:
- For \( (x, y) = (-3, -5) \):
\[ xy = -3 \times -5 = 15 \]
- For \( (x, y) = (5, 3) \):
\[ xy = 5 \times 3 = 15 \]

Thus, in both cases, the product \( xy \) is:
\[ \boxed{15} \]