1) \( p^{2}+14 p-38=0 \)

To solve the quadratic equation \( p^{2}+14p-38=0 \), we can use the quadratic formula \( p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = 14 \), and \( c = -38 \). Plugging in those values, we get \( p = \frac{-14 \pm \sqrt{14^2 - 4(1)(-38)}}{2(1)} \). This simplifies to \( p = \frac{-14 \pm \sqrt{196 + 152}}{2} = \frac{-14 \pm \sqrt{348}}{2} \). Simplifying further gives us the solutions for \( p \). Finding the square root and simplifying leads to the exact values for \( p \); don't forget to calculate the two possible values for the "±" part of the equation! Exploring solutions to quadratic equations can lead to a "eureka" moment, especially if you find ways to relate it to real-life scenarios. Think of moments when balancing a budget or maximizing profits – those scenarios often have a quadratic nature! And while solving, keep in mind the common mistake of forgetting the negative sign in front of the discriminant; this can flip your solutions on their head!