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Question
Solving the following quadratic equation by factorization method:
\[x^2 - \left( 2\sqrt{3} + 3i \right) x + 6\sqrt{3}i = 0\]
Solution
\[ x^2 - \left( 2\sqrt{3} + 3i \right) x + 6\sqrt{3}i = 0\]
\[ \Rightarrow x^2 - 2\sqrt{3}x - 3i x + 6\sqrt{3}i = 0\]
\[ \Rightarrow x\left( x - 2\sqrt{3} \right) - 3i\left( x - 2\sqrt{3} \right) = 0\]
\[ \Rightarrow \left( x - 2\sqrt{3} \right)\left( x - 3i \right) = 0\]
\[ \Rightarrow \left( x - 2\sqrt{3} \right) = 0\text { or } \left( x - 3i \right) = 0\]
\[ \Rightarrow x = 2\sqrt{3}, 3i\]
\[\text { So, the roots of the given quadratic equation are } 2\sqrt{3} \text { and } 3i . \]
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