In the following, determine whether the given quadratic equations have real roots and if so, find the roots:

$2x^2+5\sqrt3 x+6=0$

Given:

Given quadratic equation is $2x^2+5\sqrt3 x+6=0$.

To do:

We have to determine whether the given quadratic equation has real roots.

Solution:

Comparing the given quadratic equation with the standard form of the quadratic equation $ax^2+bx+c=0$, we get,

$a=2, b=5\sqrt3$ and $c=6$.

The discriminant of the standard form of the quadratic equation $ax^2+bx+c=0$ is

$D=b^2-4ac$.

Therefore,

$D=(5\sqrt3)^2-4(2)(6)=25(3)-48=75-48=27$.

As $D>0$, the given quadratic equation has real roots and the roots are

$x=\frac{-b\pm \sqrt{D}}{2a}$

$x=\frac{-5\sqrt3\pm \sqrt{27}}{2(2)}$ 

$x=\frac{-5\sqrt3\pm 3\sqrt3}{4}$ 

$x=\frac{-5\sqrt3+3\sqrt3}{4}$ or $x=\frac{-5\sqrt3-3\sqrt3}{4}$

$x=\frac{-2\sqrt3}{4}$ or $x=\frac{-8\sqrt3}{4}$

$x=\frac{-\sqrt3}{2}$ or $x=-2\sqrt3$

$x=-\frac{\sqrt3}{2}$ or $x=-2\sqrt3$

The roots are $-\frac{\sqrt3}{2}$ and $-2\sqrt3$. 

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Updated on: 10-Oct-2022

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