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Question
Solve the following quadratic equations by factorization: \[\sqrt{3} x^2 - 2\sqrt{2}x - 2\sqrt{3} = 0\]
Solution
\[\sqrt{3} x^2 - 2\sqrt{2}x - 2\sqrt{3} = 0\]
\[ \Rightarrow \sqrt{3} x^2 - 3\sqrt{2}x + \sqrt{2}x - 2\sqrt{3} = 0\]
\[ \Rightarrow \sqrt{3}x\left( x - \sqrt{6} \right) + \sqrt{2}\left( x - \sqrt{6} \right) = 0\]
\[ \Rightarrow \left( \sqrt{3}x + \sqrt{2} \right)\left( x - \sqrt{6} \right) = 0\]
\[ \Rightarrow \sqrt{3}x + \sqrt{2} = 0 \text { or } x - \sqrt{6} = 0\]
\[ \Rightarrow x = - \sqrt{\frac{2}{3}} \text { or } x = \sqrt{6}\]
Hence, the factors are \[\sqrt{6}\] and \[- \sqrt{\frac{2}{3}}\].
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