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:

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

 

Solution:

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

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

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

$\therefore$ Discriminant, $D=b^{2}-4 a c=(-5)^{2}-4 \times 2 \sqrt{3} \times \sqrt{3}=25-24=1>0$

So, the given equation has real roots.

Now, $\sqrt{D}=\sqrt{1}=1$

$\therefore \alpha=\frac{-b+\sqrt{D}}{2 a}=\frac{-(-5)+1}{2 \times 2 \sqrt{3}}=\frac{6}{4 \sqrt{3}}=\frac{\sqrt{3}}{2}$

$\beta=\frac{-b-\sqrt{D}}{2 a}=\frac{-(-5)-1}{2 \times 2 \sqrt{3}}=\frac{4}{4 \sqrt{3}}=\frac{\sqrt{3}}{3}$

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

 

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