ind the root of the equation \( x^{2}-6 x+18= \)

To solve the equation \( x^{2} - 6x + 18 = 0 \), we can use the quadratic formula, which is given by \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Here, \( a = 1 \), \( b = -6 \), and \( c = 18 \). Plugging these values into the formula gives us \( x = \frac{6 \pm \sqrt{(-6)^2 - 4 \cdot 1 \cdot 18}}{2 \cdot 1} \), which simplifies to \( x = \frac{6 \pm \sqrt{36 - 72}}{2} = \frac{6 \pm \sqrt{-36}}{2} = \frac{6 \pm 6i}{2} = 3 \pm 3i \). So the roots of the equation are \( 3 + 3i \) and \( 3 - 3i \). Imagine being a detective but for math! Quadratic equations often yield complex roots when the discriminant (the \( b^2 - 4ac \) part) is negative, just like missing clues in a case. So when you see a negative discriminant, don’t panic! Instead, embrace the imaginary number world, where every problem has a solution, even if it’s not on the real number line. While it might seem daunting at first, working with complex numbers can actually be quite thrilling! They play a crucial role in fields like electrical engineering and quantum physics, where they help to model systems with oscillations and wave functions. So don't shy away from the imaginary—it's where a whole new dimension of math opens up, making complex analysis both essential and exciting!