. \( r^{2}-9 t^{2} \)

The expression \( r^{2}-9 t^{2} \) is a classic example of a difference of squares, which can be factored using the identity \( a^{2}-b^{2} = (a-b)(a+b) \). Here, you can consider \( r^2 \) as \( a^2 \) and \( (3t)^2 \) as \( b^2 \). Thus, it factors beautifully into \( (r - 3t)(r + 3t) \), showcasing how differences of squares can help simplify or solve equations. In the real world, this factorization is powerful in various fields like physics, engineering, and finance, where quadratic equations frequently surface. For instance, when analyzing the motion of projectiles, expressions similar to \( r^{2}-9t^{2} \) can model the trajectories, allowing scientists and engineers to derive crucial insights about speeds, distances, or time intervals. Learning to factor such expressions can sharpen problem-solving skills and make complex theories more digestible!