If `z=(1+2i)/(2-i)-(2-i)/(1+2i),` then what is the value of `z^(2)+zbarz`
`(i=sqrt(-1))`

Options

A. 0
B. `-1`
C. 1
D. 8

Explanation

To solve the problem, we need to find the value of \( z^2 + \overline{z} \) where \( z = \frac{1 + 2i}{2 - i} - \frac{2 - i}{1 + 2i} \).

### Step 1: Simplifying \( z \)

First, we will simplify \( z \):

\[ z = \frac{1 + 2i}{2 - i} - \frac{2 - i}{1 + 2i} \]

To combine these fractions, we need a common denominator. The common denominator will be \( (2 - i)(1 + 2i) \).

### Step 2: Finding the first fraction

Calculating the first fraction:

\[ \frac{1 + 2i}{2 - i} \cdot \frac{1 + 2i}{1 + 2i} = \frac{(1 + 2i)(1 + 2i)}{(2 - i)(1 + 2i)} \]

Calculating the numerator:

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

Calculating the denominator:

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

Thus, the first fraction simplifies to:

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

### Step 3: Finding the second fraction

Now, calculating the second fraction:

\[ \frac{2 - i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i} = \frac{(2 - i)(1 - 2i)}{(1 + 2i)(1 - 2i)} \]

Calculating the numerator:

\[ (2 - i)(1 - 2i) = 2 - 4i - i + 2i^2 = 2 - 5i - 2 = 0 - 5i = -5i \]

Calculating the denominator:

\[ (1 + 2i)(1 - 2i) = 1 - 4i^2 = 1 + 4 = 5 \]

Thus, the second fraction simplifies to:

\[ \frac{-5i}{5} = -i \]

### Step 4: Combine the fractions

Now we can combine the two fractions:

\[ z = \frac{-3 + 4i}{4 + 3i} + i \]

To combine, we rewrite \( i \) as \( \frac{0 + 5i}{5} \):

\[ z = \frac{-3 + 4i + 0 + 5i}{4 + 3i} = \frac{-3 + 9i}{4 + 3i} \]

### Step 5: Rationalizing the denominator

Now we rationalize the denominator:

\[ z = \frac{(-3 + 9i)(4 - 3i)}{(4 + 3i)(4 - 3i)} \]

Calculating the denominator:

\[ (4 + 3i)(4 - 3i) = 16 + 9 = 25 \]

Calculating the numerator:

\[ (-3 + 9i)(4 - 3i) = -12 + 9 \cdot 4i + 3 \cdot 9 = -12 + 36i + 27 = 15 + 36i \]

Thus, we have:

\[ z = \frac{15 + 36i}{25} = \frac{15}{25} + \frac{36}{25}i = \frac{3}{5} + \frac{36}{25}i \]

### Step 6: Finding \( z^2 + \overline{z} \)

Now, we need to find \( z^2 \) and \( \overline{z} \):

1. **Finding \( z^2 \)**:

\[ z^2 = \left(\frac{3}{5} + \frac{36}{25}i\right)^2 = \left(\frac{3}{5}\right)^2 + 2 \cdot \frac{3}{5} \cdot \frac{36}{25}i + \left(\frac{36}{25}i\right)^2 \]

Calculating:

\[ = \frac{9}{25} + \frac{216}{125}i - \frac{1296}{625} \]

2. **Finding \( \overline{z} \)**:

\[ \overline{z} = \frac{3}{5} - \frac{36}{25}i \]

### Step 7: Adding \( z^2 + \overline{z} \)

Now we can find \( z^2 + \overline{z} \):

\[ z^2 + \overline{z} = \left(\frac{9}{25} - \frac{1296}{625}\right) + \left(\frac{216}{125}i - \frac{36}{25}i\right) \]

Combine the real parts and the imaginary parts:

Real part:

\[ \frac{9}{25} - \frac{1296}{625} = \frac{225}{625} - \frac{1296}{625} = -\frac{1071}{625} \]

Imaginary part:

\[ \frac{216}{125} - \frac{36}{25} = \frac{216}{125} - \frac{180}{125} = \frac{36}{125} \]

Thus, the final result is:

\[ z^2 + \overline{z} = -\frac{1071}{625} + \frac{36}{125}i \]