Question

Solve the following quadratic equations by factorization:

$\sqrt{3}{x}^2-2\sqrt{2}x-2\sqrt{3}=0$

Answer 1

$$\begin{gathered} \sqrt{3}{x}^2-2\sqrt{2}x-2\sqrt{3}=0 \\ \Rightarrow \sqrt{3}{x}^2-3\sqrt{2}x+\sqrt{2}x-2\sqrt{3}=0 \\ \Rightarrow \sqrt{3}x\left(x-\sqrt{6}\right)+\sqrt{2}\left(x-\sqrt{6}\right)=0 \\ \Rightarrow \left(\sqrt{3}x+\sqrt{2}\right)\left(x-\sqrt{6}\right)=0 \\ \Rightarrow \sqrt{3}x+\sqrt{2}=0\mathrm{or}x-\sqrt{6}=0 \\ \Rightarrow x=-\sqrt{\frac{2}{3}}\mathrm{or}x=\sqrt{6} \end{gathered}$$

Hence, the factors are $\sqrt{6}$ and $-\sqrt{\frac{2}{3}}$.