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

$4 \sqrt{3} x^{2} + 5 x - 2 \sqrt{3} = 0$ [CBSE 2013]

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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 = $- 2 \sqrt{3}$

∴ 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, $\sqrt{D} = \sqrt{121} = 11$

$∴ α = \frac{- b + \sqrt{D}}{2 a} = \frac{- 5 + 11}{2 \times 4 \sqrt{3}} = \frac{6}{8 \sqrt{3}} = \frac{\sqrt{3}}{4} β = \frac{- b - \sqrt{D}}{2 a} = \frac{- 5 - 11}{2 \times 4 \sqrt{3}} = \frac{- 16}{8 \sqrt{3}} = - \frac{2 \sqrt{3}}{3}$

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

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