To solve the quadratic equation \( x^2 - 2x + 6 = 0 \), we can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In this case, \( a = 1 \), \( b = -2 \), and \( c = 6 \).
First, we calculate the discriminant \( b^2 - 4ac \):
\[ b^2 - 4ac = (-2)^2 - 4(1)(6) = 4 - 24 = -20 \]
Since the discriminant is negative (\( -20 \)), this means there are no real solutions to the equation.
Therefore, the correct answer is:
no real solution.