Find the roots of each of the following equations, if they exist, by applying the quadratic formula:
Question:

Find the roots of each of the following equations, if they exist, by applying the quadratic formula:

$4 \sqrt{3} x^{2}+5 x-2 \sqrt{3}=0$

Solution:

The given equation is $4 \sqrt{3} x^{2}+5 x-2 \sqrt{3}=0$.

Comparing it with $a x^{2}+b x+c=0$, we get

$a=4 \sqrt{3}, b=5$ and $c=-2 \sqrt{3}$

$\therefore$ Discriminant, $D=b^{2}-4 a c=5^{2}-4 \times 4 \sqrt{3} \times(-2 \sqrt{3})=25+96=121>0$

So, the given equation has real roots.

Now, $\sqrt{D}=\sqrt{121}=11$

$\therefore \alpha=\frac{-b+\sqrt{D}}{2 a}=\frac{-5+11}{2 \times 4 \sqrt{3}}=\frac{6}{8 \sqrt{3}}=\frac{\sqrt{3}}{4}$

$\beta=\frac{-b-\sqrt{D}}{2 a}=\frac{-5-11}{2 \times 4 \sqrt{3}}=\frac{-16}{8 \sqrt{3}}=-\frac{2 \sqrt{3}}{3}$

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