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

$2 x^{2} + 5 \sqrt{3} x + 6 = 0$

Open in App

Solution

The given equation is $2 x^{2} + 5 \sqrt{3} x + 6 = 0$ .

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

a = 2, b = $5 \sqrt{3}$ and c = 6

∴ Discriminant, D = $b^{2} - 4 a c = {(5 \sqrt{3})}^{2} - 4 \times 2 \times 6 = 75 - 48 = 27 > 0$

So, the given equation has real roots.

Now, $\sqrt{D} = \sqrt{27} = 3 \sqrt{3}$

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

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

Suggest Corrections

Similar questions

Find the roots of each of the following equations, if they exist, by applying the quadratic formula:
$x^{2} - 6 x + 4 = 0$

Find the roots of each of the following equations, if they exist, by applying the quadratic formula:
$x^{2} + 5 x - (a^{2} + a - 6) = 0$

x^{2} + x + 2 = 0

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

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

Join BYJU'S Learning Program