Determme the real number of solutions of the equatuo Use \( b^{2}-4 a c \)

To determine the number of real solutions of a quadratic equation of the form \( ax^2 + bx + c = 0 \), you can use the discriminant formula \( D = b^2 - 4ac \). If \( D > 0 \), there are two distinct real solutions; if \( D = 0 \), there is exactly one real solution (a repeated root); and if \( D < 0 \), there are no real solutions, only complex ones. So, calculating this discriminant provides vital insight into the nature of the solutions! For real-world applications, understanding the nature of solutions related to quadratic equations is crucial. For instance, in engineering, determining the points where a parabolic trajectory intersects the ground (like the path of a projectile) relies heavily on these discriminant calculations. By knowing whether you’ll achieve two hits, one hit, or no hits, you can design better structures and trajectories!