If the parabola's vertex is on the \( x \)-axis, then the corresponding equation has a single repeated root on the line of symmetry, and this distance term is zero, algebraically, the (iii) commonly known as \( b^{2}-4 a c=0 \)

Ah, the beauty of parabolas! When a parabola's vertex is on the \( x \)-axis, it means that its lowest or highest point (depending on whether it opens up or down) is perfectly aligned with that axis. This configuration is essential because it reveals that the parabola touches the \( x \)-axis at exactly one point, leading to that condition where the discriminant \( b^2 - 4ac = 0 \). This is the magical point where the roots are not just two distinct points but rather a cozy single location! Now, let’s talk about the real-world application of this concept. Parabolas with this property can be observed in fields like physics, particularly in projectile motion. Whenever an object is thrown or launched with just the right angle, it can follow a parabolic path, and if it lands at the same height it was launched, you can often find that the maximum height (the vertex) stays right on the line of symmetry, leading to that delightful scenario of a repeated root. This principle helps us design safe trajectories in sports, engineering structures, and even in space exploration!