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Question
Express the following complex number in the standard form a + i b:
\[\frac{3 - 4i}{(4 - 2i)(1 + i)}\]
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Solution
\[\frac{3 - 4i}{\left( 4 - 2i \right)\left( 1 + i \right)}\]
\[ = \frac{3 - 4i}{4 + 2i - 2 i^2} \left( \because i^2 = - 1 \right)\]
\[ = \frac{3 - 4i}{6 + 2i}\]
\[ = \frac{3 - 4i}{6 + 2i} \times \frac{6 - 2i}{6 - 2i}\]
\[ = \frac{18 - 6i - 24i + 8 i^2}{36 - 4 i^2}\]
\[ = \frac{18 - 30i - 8}{36 + 4} \]
\[ = \frac{10 - 30i}{40}\]
\[ = \frac{1}{4} - \frac{3}{4}i\]
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