Find the conjugates of the following complex numbers: (3-i)^2 2+i - Vidyadip

Question

Find the conjugates of the following complex numbers: $\frac{(3-\text{i})^2}{2+\text{i}}$

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Answer

Let $\text{z}=\frac{(3-\text{i})^2}{2+\text{i}}$ $=\frac{3^2+\text{i}^2-2\times3\times\text{i}}{2+\text{i}}$ $=\frac{9-1-6\text{i}}{2+\text{i}}$ $=\frac{8-6\text{i}}{2+\text{i}}$ $=\frac{8-6\text{i}}{2+1}\times\frac{2-\text{i}}{2-\text{i}}$ $=\frac{8(2-\text{i})-6\text{i}(2-\text{i})}{2^2+1^2}$ $=\frac{16-8\text{i}-12\text{i}-6}{4+1}$ $=\frac{10-20\text{i}}{5}$ $\Rightarrow\text{z}=2-4\text{i}$ Hence $\bar{\text{z}}=2+4\text{i}$

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