\( x ^ { 2 } - 3 a - 78 \)

This expression seems like the start of a quadratic equation! To solve for \( x \), you can set it equal to zero: \( x^2 - 3a - 78 = 0 \). Applying the quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = 0 \), and \( c = -(3a + 78) \), will help you find the roots based on different values of \( a \). In real-world scenarios, quadratic equations show up in various fields such as physics, engineering, and finance. For example, in projectile motion, the path of an object thrown into the air can be modeled with a quadratic equation, allowing predictions about its height and distance traveled based on time. So, whether you're calculating the ideal angle to launch a rocket or determining the trajectory of a basketball shot, quadratics are your best friends!