Question

Solve each of the following equations by using the method of completing the square:

$\sqrt{3}{x}^2+10x+7\sqrt{3}=0$

Answer 1

$$\begin{gathered} \sqrt{3}{x}^2+10x+7\sqrt{3}=0 \\ \Rightarrow 3x^2+10\sqrt{3}x+21=0\left(\mathrm{Multiplying}\mathrm{both}\mathrm{sides}\mathrm{by}\sqrt{3}\right) \\ \Rightarrow 3x^2+10\sqrt{3}x=-21 \\ \Rightarrow {\left(\sqrt{3}x\right)}^2+2\times \sqrt{3}x\times 5+5^2=-21+5^2\left(\mathrm{Adding}{5}^2\mathrm{on}\mathrm{both}\mathrm{sides}\right) \end{gathered}$$
$$\begin{gathered} \Rightarrow {\left(\sqrt{3}x+5\right)}^2=-21+25=4=2^2 \\ \Rightarrow \sqrt{3}x+5=\pm 2\left(\mathrm{Taking}\mathrm{square}\mathrm{root}\mathrm{on}\mathrm{both}\mathrm{sides}\right) \\ \Rightarrow \sqrt{3}x+5=2\mathrm{or}\sqrt{3}x+5=-2 \\ \Rightarrow \sqrt{3}x=-3\mathrm{or}\sqrt{3}x=-7 \end{gathered}$$
$\Rightarrow x=-\frac{3}{\sqrt{3}}=-\sqrt{3}\mathrm{or}x=-\frac{7}{\sqrt{3}}=-\frac{7\sqrt{3}}{3}$
Hence, $-\sqrt{3}$ and $-\frac{7\sqrt{3}}{3}$ are the roots of the given equation.