Solve each of the following quadratic equations:

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

$\therefore \sqrt{7} x^{2}-6 x-13 \sqrt{7}=0$

$\Rightarrow \sqrt{7} x^{2}+7 x-13 x-13 \sqrt{7}=0$

$\Rightarrow \sqrt{7} x(x+\sqrt{7})-13(x+\sqrt{7})=0$

$\Rightarrow(x+\sqrt{7})(\sqrt{7} x-13)=0$

$\Rightarrow x+\sqrt{7}=0$ or $\sqrt{7} x-13=0$

$\Rightarrow x=-\sqrt{7}$ or $x=\frac{13}{\sqrt{7}}=\frac{13 \sqrt{7}}{7}$

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