If the real numbers a, b, c are distinct, then the roots of the equation $\left({\mathrm{c}}^2-\mathrm{ab}\right){\mathrm{x}}^2-2\left({\mathrm{a}}^2-\mathrm{ac}\right)\mathrm{x}+\left({\mathrm{b}}^2-\mathrm{ac}\right)=0$ will be equal, if

The given equation is $\left({\mathrm{c}}^2-\mathrm{ab}\right){\mathrm{x}}^2-2\left({\mathrm{a}}^2-\mathrm{ac}\right)\mathrm{x}+\left({\mathrm{b}}^2-\mathrm{ac}\right)=0$

Given that the equation has equal roots 

hence the discriminant is zero

it implies that 

$\begin{array}{l}\mathrm{\Delta }=0\Rightarrow \mathrm{a}\left({\mathrm{a}}^3+{\mathrm{b}}^3+{\mathrm{c}}^3-3\mathrm{abc}\right)=0\\ \Rightarrow \mathrm{a}=0\left(\mathrm{cr}\right)\mathrm{a}+\mathrm{b}+\mathrm{c}=0\text{ But not }{\mathrm{a}}^2+{\mathrm{b}}^2+{\mathrm{c}}^2\end{array}$
because a, b, c are all distinct.