The number of solutions of ` z^(2) + 2 bar(z) = 0` is

Options

A. 4
B. 3
C. 2
D. 5

Answer

To solve the equation \( z^2 + 2\bar{z} = 0 \) and find the number of solutions, we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ z^2 + 2\bar{z} = 0 \]
We can express \( \bar{z} \) in terms of \( z \). Recall that \( \bar{z} = \frac{1}{z} \) when \( z \) is not zero. However, we will first manipulate the equation directly.

### Step 2: Take the conjugate of both sides
Taking the conjugate of the entire equation gives us:
\[ \overline{z^2 + 2\bar{z}} = \overline{0} \]
This simplifies to:
\[ \bar{z}^2 + 2z = 0 \]

### Step 3: Express \(\bar{z}\) in terms of \(z\)
From the original equation, we can express \( \bar{z} \):
\[ \bar{z} = -\frac{z^2}{2} \]
Now substitute this expression for \( \bar{z} \) into the conjugate equation:
\[ \left(-\frac{z^2}{2}\right)^2 + 2z = 0 \]

### Step 4: Simplify the equation
Expanding the equation gives:
\[ \frac{z^4}{4} + 2z = 0 \]
Multiplying through by 4 to eliminate the fraction:
\[ z^4 + 8z = 0 \]

### Step 5: Factor the equation
Factoring out \( z \):
\[ z(z^3 + 8) = 0 \]
This gives us:
\[ z = 0 \quad \text{or} \quad z^3 + 8 = 0 \]

### Step 6: Solve for \(z\)
The first solution is:
\[ z = 0 \]
For the cubic equation \( z^3 + 8 = 0 \):
\[ z^3 = -8 \]
Taking the cube root gives:
\[ z = -2 \]
Using De Moivre's theorem, the cube roots of \(-8\) can be expressed as:
\[ z = 2 \text{cis} \left(\frac{\pi}{3} + \frac{2k\pi}{3}\right) \quad (k = 0, 1, 2) \]
This results in three distinct solutions:
1. \( z = 2 \text{cis} \left(\frac{\pi}{3}\right) = 2\left(\cos \frac{\pi}{3} + i \sin \frac{\pi}{3}\right) = 1 + i\sqrt{3} \)
2. \( z = 2 \text{cis} \left(\pi\right) = 2(-1) = -2 \)
3. \( z = 2 \text{cis} \left(\frac{5\pi}{3}\right) = 2\left(\cos \frac{5\pi}{3} + i \sin \frac{5\pi}{3}\right) = 1 - i\sqrt{3} \)

### Step 7: Count the solutions
Thus, we have:
1. \( z = 0 \)
2. \( z = 1 + i\sqrt{3} \)
3. \( z = -2 \)
4. \( z = 1 - i\sqrt{3} \)

In total, there are **4 solutions** to the equation \( z^2 + 2\bar{z} = 0 \).

### Final Answer:
The number of solutions is **4**.

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