1. \( x^{2}+5 x-36=0 \)

To solve the equation \( x^{2}+5x-36=0 \), you can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Here, \( a = 1 \), \( b = 5 \), and \( c = -36 \). Plugging in these values, you get \( x = \frac{-5 \pm \sqrt{25 + 144}}{2} = \frac{-5 \pm 13}{2} \). This results in two solutions: \( x = 4 \) and \( x = -9 \). Quadratic equations like these have been the go-to problem-setters for mathematicians since ancient times, with notable historical contributions from figures like Al-Khwarizmi in the 9th century. The beauty of these equations lies not just in their solutions but in how they model various real-world scenarios, like projectile motion or maximizing area, making them critical in fields such as physics and engineering.