Interval that the sum of the roots of a quadratic equation can be in.

Question

I am given the equation:

$2x^2-2mx+m^2-2m=0$

Where $m\in \mathbb{R}$; $x_1$ and $x_2$ are the roots of the equation. The question asks for the interval of the sum of the roots, that is $x_1 + x_2$.

This is what I did:

Using Vieta's Formulas

$x_1 + x_2 = -\dfrac{b}{a}=-\dfrac{-2m}{2}=m.$

And since $m\in \mathbb{R}$, I thought that the interval that the sum can be in is $\mathbb{R}$. But that proved to be wrong and I don't know how to solve it, all I now know is that the true answer should be $[0,4]$.

Answer 1

You are correct that the sum of the two roots can take any value assuming that $x_1,x_2\in\mathbb{C}$. If instead we are limited to $x_1,x_2\in\mathbb{R}$ then we require a non-negative discriminant so $$b^2-4ac=(-2m)^2-4(2)(m^2-2m)\ge0\implies m\in[0,4]$$ Hence the sum of the roots $x_1+x_2=m\in[0,4]$.