Solve the equation using the quadratic formula. \[ x^{2}-2 x+26=0 \] The solution set is \{ \}. (Simplify your answer. Type an exact answer, using radicals and \( i \) as needed. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed.)

To use the quadratic formula, we start with the standard format of a quadratic equation \( ax^2 + bx + c = 0 \). For the equation \( x^2 - 2x + 26 = 0 \), we have: - \( a = 1 \) - \( b = -2 \) - \( c = 26 \) The quadratic formula is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] First, we calculate the discriminant \( b^2 - 4ac \): \[ b^2 - 4ac = (-2)^2 - 4(1)(26) = 4 - 104 = -100 \] Since the discriminant is negative, the solutions will involve imaginary numbers. Now we substitute into the quadratic formula: \[ x = \frac{-(-2) \pm \sqrt{-100}}{2(1)} = \frac{2 \pm \sqrt{-100}}{2} \] Simplifying \( \sqrt{-100} \): \[ \sqrt{-100} = 10i \] Now we can simplify the expression further: \[ x = \frac{2 \pm 10i}{2} = 1 \pm 5i \] Thus, the solution set is: \[ \{ 1 + 5i, 1 - 5i \} \]