\( 6 x ^ { 2 } + 7 x y - 3 y ^ { 2 } . \)

This expression represents a quadratic function in two variables, \(x\) and \(y\). Quadratic equations like this one can be graphed as a conic section, and in this case, it describes a hyperbola. Understanding how to graph it involves finding the critical points and interpreting the discriminant of the quadratic form, which can reveal the nature of the curves. In real-world scenarios, such equations can model a variety of phenomena, from projectile motion to areas in different geometric configurations, making them versatile tools in fields like physics, engineering, and economics. For example, an analysis of profit margins or yield optimization can involve using quadratic equations to determine optimal values.