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.