To solve the quadratic equation x^2 - 4x + 5 = 0, we can use the quadratic formula:

x = (-(-4) ± √((-4)^2 - 4(1)(5))) / 2(1)
x = (4 ± √(16 - 20)) / 2
x = (4 ± √(-4)) / 2
x = (4 ± 2i) / 2
x = 2 ± i

Therefore, the solutions to the quadratic equation x^2 - 4x + 5 = 0 are x = 2 + i and x = 2 - i.