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Question
Solve the following quadratic equation:
\[x^2 - \left( \sqrt{2} + i \right) x + \sqrt{2}i = 0\]
Solution
\[ x^2 - \left( \sqrt{2} + i \right) x + \sqrt{2} i = 0\]
\[\text { Comparing the given equation with the general form } a x^2 + bx + c = 0, \text { we get }\]
\[a = 1, b = - \left( \sqrt{2} + i \right) \text { and } c = \sqrt{2}i\]
\[x = \frac{- b \pm \sqrt{b^2 - 4ac}}{2a}\]
\[ \Rightarrow x = \frac{\left( \sqrt{2} + i \right) \pm \sqrt{\left( \sqrt{2} + i \right)^2 - 4\sqrt{2}i}}{2}\]
\[ \Rightarrow x = \frac{\left( \sqrt{2} + i \right) \pm \sqrt{1 - 2\sqrt{2} i}}{2} \]
\[ \Rightarrow x = \frac{\left( \sqrt{2} + i \right) \pm \sqrt{\left( \sqrt{2} \right)^2 - 1^2 - 2\sqrt{2} i}}{2}\]
\[ \Rightarrow x = \frac{\left( \sqrt{2} + i \right) \pm \sqrt{\left( \sqrt{2} - i \right)^2}}{2}\]
\[ \Rightarrow x = \frac{\left( \sqrt{2} + i \right) \pm \left( \sqrt{2} - i \right)}{2}\]
\[ \Rightarrow x = \sqrt{2}, i \]
\[\text { So, the roots of the given quadratic equation are } \sqrt{2} \text { and } i .\]
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