Quadratic Equations - Mixed Roots

Question

This is probably a silly question but why is it that when a quadratic equation has a single root it must be a repeated root. Why can't the second root be an imaginary root?

Answer 1

It isn't quite clear what you are asking.

If the quadratic has a single root then it must be repeated, otherwise it wouldn't have a single root.

If you are asking why a single root must be a real root then assume the opposite - that the root is imaginary. So you have a quadratic of the form $(x + u + iv)^2$, where the repeated imaginary root is $-u-iv$. But if we expand the above you get:

$$(x + u + iv)^2 = x^2 + (2u + 2iv)x + u^2-v^2+2iv$$

but that doesn't have real coefficients, so it isn't a quadratic by the normal definition.