Why can a quadratic equation not have one real and one complex root?
A quadratic equation is of the form ax^2 + bx + c = 0, where the coefficients a, b and c are real. A quadratic equation may take complex values for x but the coefficients are always real.
Now assume that a quadratic equation has one real root R and one complex root R' + C'i. We can write the equation as
(x - R)(x - (R' + C'i)) = ax^2 + bx + c
=> x^2 - (R' + C'i)x - Rx + R*(R' + C'i) = ax^2 + bx + c
=> x^2 - (R + R' + C'i)x + R*(R' + C'i) = ax^2 + bx + c
Equate the coefficients of x^2 , x and the numeric coefficients
a = 1, b = - (R + R' + C'i) and c = R*(R' + C'i)
This makes both b and c complex, which is not allowed as they have to be real.
This is the reason why if quadratic equations have complex roots, they are in pairs and form complex conjugates. That eliminates the complex parts when a, b and c are being determined.
Quadratic equations with complex roots always have them in pairs which are complex conjugate.