c) \( x^{2}-2 x-8=0 \)

To solve the equation \( x^{2}-2 x-8=0 \), you can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Here, \( a = 1 \), \( b = -2 \), and \( c = -8 \). Plugging in these values gives: \( x = \frac{2 \pm \sqrt{(-2)^2 - 4(1)(-8)}}{2(1)} \), simplifying to \( x = \frac{2 \pm \sqrt{4 + 32}}{2} \), which further results in \( x = \frac{2 \pm \sqrt{36}}{2} \). So, your roots are \( x = 5 \) and \( x = -3 \). Now, let's talk about how this kind of equation shows up in real life! Quadratic equations often pop up in scenarios involving area, projectile motion, and even economic models. For example, if you’re calculating the area of a rectangular garden where one side length is dependent on the other, you might end up with a quadratic equation to solve for the dimensions. They help us understand how changes in one variable can drastically affect another! An important tip when solving quadratics is to always double-check your factors if you're factoring the equation instead of using the quadratic formula! It’s easy to make little mistakes, especially with signs and coefficients. A common mistake is overlooking the negative signs in a quadratic equation, which could change the roots altogether. Keep a calculator handy to verify your solutions and avoid turning a simple problem into a mathematical minefield!