d) \( x^{2}-26 x+164=0 \)

To solve the quadratic equation \( x^{2}-26x+164=0 \), we can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Here, \( a = 1 \), \( b = -26 \), and \( c = 164 \). Plugging these into the formula gives us \( x = \frac{26 \pm \sqrt{(-26)^2 - 4 \cdot 1 \cdot 164}}{2 \cdot 1} = \frac{26 \pm \sqrt{676 - 656}}{2} = \frac{26 \pm \sqrt{20}}{2} = \frac{26 \pm 2\sqrt{5}}{2} = 13 \pm \sqrt{5} \). Therefore, the solutions are \( x = 13 + \sqrt{5} \) and \( x = 13 - \sqrt{5} \). Many mathematicians have found that the quadratic formula is not just useful for textbooks; it has real-world applications in fields like engineering and finance. For example, optimizing areas under curves and maximizing profit margins are scenarios where this formula comes in handy. It's a key tool that helps professionals make informed decisions based on mathematical predictions! However, when applying the quadratic formula, it's easy to trip up by miscalculating the discriminant, \( b^2 - 4ac \). Double-check your arithmetic here! Another common mistake is forgetting to simplify square roots or fractions after substitution; doing so keeps your answers neat and easier to interpret. Always remember, clear fronts can lead to a clearer understanding!