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Question
Find the conjugate of the following complex number:
\[\frac{(1 + i)(2 + i)}{3 + i}\]
Solution
\[\text { Let} z = \frac{\left( 1 + i \right)\left( 2 + i \right)}{3 + i}\]
\[ = \frac{2 + i + 2i + i^2}{3 + i}\]
\[ = \frac{1 + 3i}{3 + i}\]
\[ = \frac{1 + 3i}{3 + i} \times \frac{3 - i}{3 - i}\]
\[ = \frac{3 - i + 9i - 3 i^2}{9 - i^2}\]
\[ = \frac{6 + 8i}{10}\]
\[ = \frac{3 + 4i}{5}\]
\[ \Rightarrow \bar{z} = \frac{3 - 4i}{5}\]
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