$\text{ If }a+b+c=0,\text{ then the roots of the equation }\left(c^2-ab\right)x^2-2\left(a^2-bc\right)x+\left(b^2-ac\right)=0\text{ are }$

$\begin{array}{l}\mathrm{D}=4\left[{\left({\mathrm{a}}^2-\mathrm{bc}\right)}^2-\left({\mathrm{b}}^2-\mathrm{ac}\right)\left({\mathrm{c}}^2-\mathrm{ab}\right)\right]\\ =4\left[{\mathrm{a}}^4+{\mathrm{b}}^2{\mathrm{c}}^2-2{\mathrm{a}}^2\mathrm{bc}-\left\{{\mathrm{b}}^2{\mathrm{c}}^2-{\mathrm{ac}}^3-{\mathrm{ab}}^3+{\mathrm{a}}^2\mathrm{bc}\right\}\right]\\ =4\mathrm{a}\left({\mathrm{a}}^3+{\mathrm{b}}^3+{\mathrm{c}}^3-3\mathrm{abc}\right)\\ \mathrm{D}=4\mathrm{a}(\mathrm{a}+\mathrm{b}+\mathrm{c})\left({\mathrm{a}}^2+{\mathrm{b}}^2+{\mathrm{b}}^2-\mathrm{ab}-\mathrm{bc}-\mathrm{ac}\right)=0\\ [ \mathrm{as} \mathrm{a}+\mathrm{b}+\mathrm{c}=0]\end{array}$

Hence roots are real and distinct.