10. \( x^{2}-6 x-4 \)

To solve the quadratic equation \( x^{2}-6 x-4 = 0 \) using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), we identify \( a = 1 \), \( b = -6 \), and \( c = -4 \). Plugging these values into the formula gives us \( x = \frac{6 \pm \sqrt{(-6)^2 - 4 \cdot 1 \cdot (-4)}}{2 \cdot 1} \). Simplifying this results in \( x = \frac{6 \pm \sqrt{36 + 16}}{2} \), which simplifies to \( x = \frac{6 \pm \sqrt{52}}{2} \), or more simply \( x = 3 \pm \sqrt{13} \). A fun fact about quadratic equations is that they can often represent various real-world scenarios! Whether it's calculating projectile motion in physics or optimizing profit in economics, these equations pop up everywhere—often in ways we don’t even realize we’re using them. Next time you see a bridge's design or a ball’s trajectory, remember, quadratics might be the secret sauce behind that engineering magic. You might want to avoid some common pitfalls while solving quadratics. One frequent mistake is accidentally miscalculating the discriminant, the part under the square root! Double-checking this part can save you from wrong roots. Additionally, if factoring, remember to consider both signs in your search for factors; they can lead you down different paths. Happy solving!