Question

Solve each of the following quadratic equations:

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

Answer 1

We write, $10x=3x+7x$ as $\sqrt{3}{x}^2\times 7\sqrt{3}=21x^2=3x\times 7x$

$$\begin{gathered} \therefore \sqrt{3}{x}^2+10x+7\sqrt{3}=0 \\ \Rightarrow \sqrt{3}{x}^2+3x+7x+7\sqrt{3}=0 \\ \Rightarrow \sqrt{3}x\left(x+\sqrt{3}\right)+7\left(x+\sqrt{3}\right)=0 \\ \Rightarrow \left(x+\sqrt{3}\right)\left(\sqrt{3}x+7\right)=0 \end{gathered}$$
$$\begin{gathered} \Rightarrow x+\sqrt{3}=0\mathrm{or}\sqrt{3}x+7=0 \\ \Rightarrow x=-\sqrt{3}\mathrm{or}x=-\frac{7}{\sqrt{3}}=-\frac{7\sqrt{3}}{3} \end{gathered}$$
Hence, the roots of the given equation are $-\sqrt{3}$ and $-\frac{7\sqrt{3}}{3}$.