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 \((1 + 2i)^2\).

Using the formula \((a + b)^2 = a^2 + 2ab + b^2\):
\[ (1 + 2i)^2 = 1^2 + 2 \cdot 1 \cdot 2i + (2i)^2 \]
Calculating each term:
- \(1^2 = 1\)
- \(2 \cdot 1 \cdot 2i = 4i\)
- \((2i)^2 = 4i^2 = 4(-1) = -4\)

Combining these:
\[ (1 + 2i)^2 = 1 + 4i - 4 = -3 + 4i \]

### Step 2: Rewrite the expression
Now we can rewrite the original expression:
\[ \frac{(1 + 2i)^2}{3 - i} = \frac{-3 + 4i}{3 - i} \]

### Step 3: Rationalize the denominator
To rationalize the denominator, multiply the numerator and denominator by the conjugate of the denominator, which is \(3 + i\):
\[ \frac{(-3 + 4i)(3 + i)}{(3 - i)(3 + i)} \]

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

Calculating the numerator:
\[ (-3 + 4i)(3 + i) = -3 \cdot 3 - 3 \cdot i + 4i \cdot 3 + 4i \cdot i = -9 - 3i + 12i + 4i^2 \]
Since \(i^2 = -1\):
\[ 4i^2 = 4(-1) = -4 \]
So,
\[ -9 - 3i + 12i - 4 = -9 + 9i - 4 = -13 + 9i \]

Now, we can write the expression:
\[ \frac{-13 + 9i}{10} \]

### Step 4: Separate real and imaginary parts
This can be expressed as:
\[ -\frac{13}{10} + \frac{9}{10}i \]

### Step 5: Find the conjugate
The conjugate of a complex number \(x + yi\) is \(x - yi\). Therefore, the conjugate of \(-\frac{13}{10} + \frac{9}{10}i\) is:
\[ -\frac{13}{10} - \frac{9}{10}i \]

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

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