b) \( x^{2}+9 x-36 \)

To solve the quadratic equation \( x^{2}+9x-36 = 0 \) using the quadratic formula, you can substitute \( a = 1 \), \( b = 9 \), and \( c = -36 \) into the formula \( x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \). This yields two potential solutions: \( x = \frac{{-9 + 15}}{2} = 3 \) and \( x = \frac{{-9 - 15}}{2} = -12 \). So, your solutions are \( x = 3 \) and \( x = -12 \)! Did you know that the quadratic formula provides a universal method for solving quadratics, regardless of their complexity? It was popularized by the ancient Greeks and has been mathematically refined over centuries. So the next time you're solving a quadratic, remember this hefty mathematical history behind that nifty formula! In practice, quadratics are everywhere—from physics equations to business profit models. For instance, you can use quadratics to determine maximum profit in a given pricing strategy or to model the path of a ball thrown in the air. This real-world application shows just how integral these mathematical concepts are to everyday life!