Solve each equation by completing the square. 1) \( p^{2}+14 p-38=0 \) 3) \( a^{2}+14 a-51=0 \) 5) \( x^{2}+6 x+8=0 \) 7) \( x^{2}+14 x-15=0 \) 15

Completing the square is not just a method for solving quadratic equations—it’s like uncovering the hidden beauty of parabolas! When you rearrange equations into a perfect square trinomial, you're essentially expressing the quadratic in a way that makes zeros explicit. It also shines a light on the vertex form of a parabola, crucial in graphing and understanding its behavior. In real-world situations, the concept of completing the square comes in handy more often than you might think! For instance, if you are designing a parabolic arch in architecture or optimizing the area under a projectile's flight path, transforming quadratic equations into vertex form can make your calculations cleaner and more intuitive. So, next time you're crafting a design or figuring out physics problems, don’t forget about the magic of completing the square!