Solve each of the following quadratic equations:
Question:

Solve each of the following quadratic equations:

$\sqrt{3} x^{2}+10 x-8 \sqrt{3}=0$

Solution:

Consider $\sqrt{3} x^{2}+10 x-8 \sqrt{3}=0$

Factorising by splitting the middle term;

$\sqrt{3} x^{2}+12 x-2 x-8 \sqrt{3}=0$

$\Rightarrow \sqrt{3} x(x+4 \sqrt{3})-2(x+4 \sqrt{3})=0$

$\Rightarrow(\sqrt{3} x-2)(x+4 \sqrt{3})=0$

$\Rightarrow \sqrt{3} x-2=0$ or $x+4 \sqrt{3}=0$

$\Rightarrow x=\frac{2}{\sqrt{3}}$ or $x=-4 \sqrt{3}$

Hence, the roots of the given equation are $\frac{2}{\sqrt{3}}$ and $-4 \sqrt{3}$.