d) \( x^{2}-6 x-16=0 \)

To solve the quadratic equation \( x^{2}-6x-16=0 \), you can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Here, \( a = 1 \), \( b = -6 \), and \( c = -16 \). Plugging in the values gives you \( x = \frac{6 \pm \sqrt{(-6)^2 - 4 \cdot 1 \cdot (-16)}}{2 \cdot 1} \), which simplifies to \( x = \frac{6 \pm \sqrt{36 + 64}}{2} = \frac{6 \pm \sqrt{100}}{2} = \frac{6 \pm 10}{2} \). Thus, the solutions are \( x = 8 \) and \( x = -2 \). Did you know that quadratic equations have been used since ancient civilizations like the Babylonians, who could solve them geometrically? Their methods laid the groundwork for algebra as we know it today! It's fascinating how these mathematical puzzles have persisted and evolved through centuries. Speaking of solving these equations, one common mistake is miscalculating the discriminant \( b^2 - 4ac \). Always double-check your arithmetic here, as a small error can lead you astray when finding your solution. Keeping your calculations organized can prevent these slip-ups!