\( x ^ { 2 } + 2 x y - y ^ { 2 } ; \quad x = 3 \)

Let’s plug \( x = 3 \) into the expression \( x^2 + 2xy - y^2 \). This gives us: \[ 3^2 + 2(3)y - y^2 = 9 + 6y - y^2 \] Now, we can rewrite the expression as: \[ -y^2 + 6y + 9 \] This is a quadratic in terms of \( y \), which opens downwards (since the coefficient of \( y^2 \) is negative). Quadratics are not only fun to explore mathematically, but they also graph out beautifully as parabolas! If you're solving or optimizing this expression, you might find it helpful to identify the vertex. The vertex of a quadratic given by \( ay^2 + by + c \) can be found using \( y = -\frac{b}{2a} \). In this case, set \( a = -1 \) and \( b = 6 \). The coordinates of the vertex will tell you the maximum point on the graph – a neat trick for visualizing your results!