\( \left. \begin{array} { l } { 8 x ^ { 2 } - 4 x } \\ { 3 x ^ { 2 } - 2 x } \\ { - 2 x ^ { 6 } - 3 x ^ { 4 } } \\ { - 2 x ^ { 3 } + 42 } \end{array} \right. \)

Did you know that quadratic functions, like the ones in your list, were extensively studied by ancient mathematicians? The roots of these equations can be traced back to the Babylonians around 2000 BC! They developed methods to solve certain quadratic equations using geometric approaches, long before the terms and symbols we use today even existed. Fast forward to the Renaissance, and you’ll find European mathematicians expanding on these ideas, leading to the modern algebra we know today. In terms of real-world applications, quadratic equations are everywhere! From projectile motion to optimizing profits in business, these equations help model various scenarios. For instance, if you're an aspiring architect, you might use quadratics to design the perfect arches and curves! Just think of it as the mathematical magic behind those stunning structures you see in the city. So whether you're calculating the trajectory of a basketball or maximizing park area, quadratics have got your back!