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

$2 \sqrt{3} x^{2} - 5 x + \sqrt{3} = 0$ [CBSE 2011]

Open in App

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 = $\sqrt{3}$

∴ 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, $\sqrt{D} = \sqrt{1} = 1$

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

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

Suggest Corrections

Similar questions

3 x^{2} - 2 \sqrt{6} x + 2 = 0

x^{2} - (\sqrt{3} + 1) x + \sqrt{3} = 0

2 x^{2} + a x - a^{2} = 0

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

Find the roots of each of the following equations, if they exist, by applying the quadratic formula:
$3 x^{2} - 2 x + 2 = 0$

Join BYJU'S Learning Program