x^2-6x+18=0

To solve the quadratic equation \(x^2 - 6x + 18 = 0\), we can use the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

In this equation, \(a = 1\), \(b = -6\), and \(c = 18\).

First, we calculate the discriminant \(b^2 - 4ac\):

\[ b^2 - 4ac = (-6)^2 - 4(1)(18) = 36 - 72 = -36 \]

Since the discriminant is negative, this equation has two complex (imaginary) solutions.

Now we can substitute the values into the quadratic formula:

\[ x = \frac{-(-6) \pm \sqrt{-36}}{2(1)} = \frac{6 \pm \sqrt{-36}}{2} = \frac{6 \pm 6i}{2} \]

Now simplify:

\[ x = \frac{6}{2} \pm \frac{6i}{2} = 3 \pm 3i \]

Thus, the solutions to the equation \(x^2 - 6x + 18 = 0\) are:

\[ x = 3 + 3i \quad \text{and} \quad x = 3 - 3i \]