iven: \( q=\sqrt{b^{2}-4 a c} \)

The expression \( q=\sqrt{b^{2}-4 a c} \) is famously known as the discriminant of a quadratic equation in the form \( ax^2 + bx + c = 0 \). This square root term provides crucial insights into the nature of the roots of the quadratic equation: if \( q > 0 \), there are two distinct real roots; if \( q = 0 \), there’s exactly one real root (a repeated root); and if \( q < 0 \), the equation has no real roots, only complex ones. In the realm of real-world applications, the discriminant helps in fields like engineering and physics by determining the stability of systems. For example, when modeling the trajectory of a projectile, assessing the potential outcomes of a design relies on understanding whether the solutions to the quadratic equation yield real points of intersection with the ground—crucial for ensuring safety and performance in designs!