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

Options

A. `(1)/(5)`
B. `(1)/(3)`
C. `-(1)/(3)`
D. none of these

Answer

To find the real part of the expression \((1 + i)^2 / (3 - i)\), we will follow these steps:

### Step-by-Step Solution:

1. **Calculate \((1 + i)^2\)**:
\[ (1 + i)^2 = 1^2 + 2 \cdot 1 \cdot i + i^2 = 1 + 2i + (-1) = 2i \]

2. **Set up the expression**:
We now have:
\[ \frac{(1 + i)^2}{3 - i} = \frac{2i}{3 - i} \]

3. **Multiply the numerator and denominator by the conjugate of the denominator**:
The conjugate of \(3 - i\) is \(3 + i\). Therefore, we multiply both the numerator and denominator by \(3 + i\):
\[ \frac{2i(3 + i)}{(3 - i)(3 + i)} \]

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

5. **Calculate the numerator**:
\[ 2i(3 + i) = 6i + 2i^2 = 6i + 2(-1) = 6i - 2 = -2 + 6i \]

6. **Combine the results**:
Now we have:
\[ \frac{-2 + 6i}{10} = \frac{-2}{10} + \frac{6i}{10} = -\frac{1}{5} + \frac{3i}{5} \]

7. **Identify the real part**:
The real part of the expression is:
\[ -\frac{1}{5} \]

### Final Answer:
The real part of \(\frac{(1 + i)^2}{3 - i}\) is \(-\frac{1}{5}\).

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