(7) Given that \( (x+y)^{2}=8 \) and \( x y=3 \), find the value of \( x^{2}+y^{2} \).

To find the value of \( x^2 + y^2 \), we can use the identity that relates the square of a sum to the individual squares. Specifically, we can start with the equation: \[ (x+y)^2 = x^2 + y^2 + 2xy \] We know from the problem that \( (x+y)^2 = 8 \) and \( xy = 3 \). Now we substitute the known values into the equation: \[ 8 = x^2 + y^2 + 2(3) \] This simplifies to: \[ 8 = x^2 + y^2 + 6 \] Subtracting 6 from both sides gives: \[ x^2 + y^2 = 8 - 6 = 2 \] Thus, the value of \( x^2 + y^2 \) is \[ \boxed{2} \]