Find the roots of the following quadratic equations (if they exist) by the method of completing the square.

$4x^2 + 4\sqrt3 x + 3 = 0$

Given:

Given quadratic equation is $4x^2 + 4\sqrt3 x + 3 = 0$.

To do:

We have to find the roots of the given quadratic equation.

Solution:

$4x^2 + 4\sqrt3 x + 3 = 0$

$4(x^2 + \sqrt3 x +\frac{3}{4})=0$  

$x^2+2\times \frac{1}{2} \times \sqrt3 x =-\frac{3}{4}$

$x^2+2\frac{\sqrt3}{2}x=-\frac{3}{4}$

Adding $(\frac{\sqrt3}{2})^2$ on both sides completes the square. Therefore,

$x^2+2\frac{\sqrt3}{2}x+(\frac{\sqrt3}{2})^2=-\frac{3}{4}+(\frac{\sqrt3}{2})^2$

$(x+\frac{\sqrt3}{2})^2=-\frac{3}{4}+\frac{3}{4}$      (Since $(a+b)^2=a^2+2ab+b^2$)

$(x+\frac{\sqrt3}{2})^2=0$

$x+\frac{\sqrt3}{2}=0$

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

The values of $x$ are $-\frac{\sqrt3}{2}$ and $-\frac{\sqrt3}{2}$.