The conjugate of ` ((1+2i)^(2))/(3-i)` is

Options

A. `(-13)/10 + 9/10 i`
B. `(-13)/10 - 9/10 i`
C. `13/10 + 9/10 i`
D. `13/10 - 9/10i`

Answer

To find the conjugate of the expression \(\frac{(1 + 2i)^2}{3 - i}\), we will follow these steps:

### Step 1: Simplify the numerator
First, we need to simplify the numerator \((1 + 2i)^2\).

\[ (1 + 2i)^2 = 1^2 + 2 \cdot 1 \cdot 2i + (2i)^2 \]
\[ = 1 + 4i + 4i^2 \]
Since \(i^2 = -1\), we have:
\[ = 1 + 4i - 4 = -3 + 4i \]

### Step 2: Rewrite the expression
Now, we can rewrite the expression as:
\[ \frac{-3 + 4i}{3 - i} \]

### Step 3: Rationalize the denominator
To simplify this further, we will rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, which is \(3 + i\).

\[ \frac{(-3 + 4i)(3 + i)}{(3 - i)(3 + i)} \]

### Step 4: Calculate the denominator
Calculating the denominator:
\[ (3 - i)(3 + i) = 3^2 - i^2 = 9 - (-1) = 10 \]

### Step 5: Calculate the numerator
Now, we calculate the numerator:
\[ (-3 + 4i)(3 + i) = -9 - 3i + 12i + 4i^2 \]
\[ = -9 + 9i + 4(-1) = -9 + 9i - 4 = -13 + 9i \]

### Step 6: Combine results
Now, we can combine the results:
\[ \frac{-13 + 9i}{10} = -\frac{13}{10} + \frac{9}{10}i \]

### Step 7: Find the conjugate
The conjugate of a complex number \(z = x + yi\) is given by \(z^* = x - yi\). Thus, the conjugate of \(-\frac{13}{10} + \frac{9}{10}i\) is:
\[ -\frac{13}{10} - \frac{9}{10}i \]

### Final Answer
The conjugate of \(\frac{(1 + 2i)^2}{3 - i}\) is:
\[ -\frac{13}{10} - \frac{9}{10}i \]

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