We write, $-6 x=7 x-13 x$ as $\sqrt{7} x^{2} \times(-13 \sqrt{7})=-91 x^{2}=7 x \times(-13 x)$

$\begin{array}{l}
\therefore \sqrt{7} x^{2}-6 x-13 \sqrt{7}=0 \\
\Rightarrow \sqrt{7} x^{2}+7 x-13 x-13 \sqrt{7}=0 \\
\Rightarrow \sqrt{7} x(x+\sqrt{7})-13(x+\sqrt{7})=0 \\
\Rightarrow(x+\sqrt{7})(\sqrt{7} x-13)=0 \\
\Rightarrow x+\sqrt{7}=0 \text { or } \sqrt{7} x-13=0 \\
\Rightarrow x=-\sqrt{7} \text { or } x=\frac{13}{\sqrt{7}}=\frac{13 \sqrt{7}}{7}
\end{array}$

Hence, the roots of the given equation are $-\sqrt{7}$ and $\frac{13 \sqrt{7}}{7}$.