Question

Find the roots of the following quadratic equations, if they exist, by the method of completing the square:
$4x^2+4\sqrt{3}x+3=0$

Answer 1

Step 1: Converting the expression into square of addition formula

Given expression $4x^2+4\sqrt{3}x+3=0$
$\Rightarrow 4x^2+4\sqrt{3}x+3=0$
$$\begin{gathered} \Rightarrow {\left(2x\right)}^2+2\times 2x\times \sqrt{3}+{\left(\sqrt{3}\right)}^2=0 \\ \Rightarrow {\left(2x+\sqrt{3}\right)}^2=0\mathbf{}\mathbf{[}\mathbf{\because }{\mathbf{(}\mathbf{a}\mathbf{+}\mathbf{b}\mathbf{)}}^{\mathbf{2}}\mathbf{=}{\mathbf{a}}^{\mathbf{2}}\mathbf{+}\mathbf{2}\mathbf{ab}\mathbf{+}{\mathbf{b}}^{\mathbf{2}}\mathbf{]} \end{gathered}$$

Step 2: Finding the roots by solving above expression

$$\begin{gathered} \Rightarrow {\left(2x+\sqrt{3}\right)}^2=0 \\ \Rightarrow \left(2x+\sqrt{3}\right)\left(2x+\sqrt{3}\right)=0 \\ \Rightarrow x=-\frac{\sqrt{3}}{2},-\frac{\sqrt{3}}{2} \end{gathered}$$

Hence the required roots are $-\frac{\sqrt{3}}{2},-\frac{\sqrt{3}}{2}$