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

Options

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

Answer

To solve the problem, we need to evaluate the expression \( z = \frac{1 + 2i}{2 - i} - \frac{2 - i}{1 + 2i} \) and then find \( z^2 + z \bar{z} \).

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

We start with the expression for \( z \):

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

To combine these two fractions, we need a common denominator. The common denominator will be the product of the two denominators:

\[ (2 - i)(1 + 2i) \]

### Step 2: Calculate the numerator

Now we calculate the numerator:

\[ \text{Numerator} = (1 + 2i)(1 + 2i) - (2 - i)(2 - i) \]

Calculating \( (1 + 2i)(1 + 2i) \):

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

Now calculating \( (2 - i)(2 - i) \):

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

Now substituting back into the numerator:

\[ \text{Numerator} = (-3 + 4i) - (3 - 4i) = -3 + 4i - 3 + 4i = -6 + 8i \]

### Step 3: Calculate the denominator

Now we calculate the denominator:

\[ \text{Denominator} = (2 - i)(1 + 2i) = 2 + 4i - i - 2 = 2 + 3i \]

### Step 4: Combine the fractions

Now we have:

\[ z = \frac{-6 + 8i}{2 + 3i} \]

To simplify this, we multiply the numerator and the denominator by the conjugate of the denominator:

\[ z = \frac{(-6 + 8i)(2 - 3i)}{(2 + 3i)(2 - 3i)} \]

Calculating the denominator:

\[ (2 + 3i)(2 - 3i) = 4 + 9 = 13 \]

Calculating the numerator:

\[ (-6 + 8i)(2 - 3i) = -12 + 18i + 16i + 24 = 12 + 34i \]

Thus,

\[ z = \frac{12 + 34i}{13} = \frac{12}{13} + \frac{34}{13}i \]

### Step 5: Calculate \( z^2 \)

Now we need to calculate \( z^2 \):

\[ z^2 = \left(\frac{12}{13} + \frac{34}{13}i\right)^2 = \left(\frac{12^2}{13^2} + 2 \cdot \frac{12}{13} \cdot \frac{34}{13}i + \left(\frac{34}{13}i\right)^2\right) \]

Calculating each term:

\[ \frac{12^2}{13^2} = \frac{144}{169}, \quad 2 \cdot \frac{12}{13} \cdot \frac{34}{13}i = \frac{816}{169}i, \quad \left(\frac{34}{13}i\right)^2 = -\frac{1156}{169} \]

So,

\[ z^2 = \frac{144 - 1156}{169} + \frac{816}{169}i = \frac{-1012}{169} + \frac{816}{169}i \]

### Step 6: Calculate \( z \bar{z} \)

Next, we calculate \( z \bar{z} \):

\[ \bar{z} = \frac{12}{13} - \frac{34}{13}i \]

Thus,

\[ z \bar{z} = \left(\frac{12}{13} + \frac{34}{13}i\right)\left(\frac{12}{13} - \frac{34}{13}i\right) = \frac{12^2 + 34^2}{13^2} = \frac{144 + 1156}{169} = \frac{1300}{169} \]

### Step 7: Combine \( z^2 + z \bar{z} \)

Finally, we combine:

\[ z^2 + z \bar{z} = \left(\frac{-1012}{169} + \frac{816}{169}i\right) + \frac{1300}{169} = \frac{-1012 + 1300}{169} + \frac{816}{169}i = \frac{288}{169} + \frac{816}{169}i \]

### Final Answer

The final value of \( z^2 + z \bar{z} \) is:

\[ \frac{288}{169} + \frac{816}{169}i \]