The number of solutions of `z^2+overlinez = 0` is

Options

A. 1
B. 2
C. 3
D. 4

Explanation

To solve the equation \( z^2 + \overline{z} = 0 \), we will follow these steps:

### Step 1: Express \( z \) in terms of its real and imaginary parts
Let \( z = x + iy \), where \( x \) and \( y \) are real numbers. The conjugate \( \overline{z} \) can be expressed as:
\[ \overline{z} = x - iy \]

### Step 2: Substitute \( z \) and \( \overline{z} \) into the equation
Substituting \( z \) and \( \overline{z} \) into the equation gives:
\[ (x + iy)^2 + (x - iy) = 0 \]

### Step 3: Expand \( z^2 \)
Now, we will expand \( (x + iy)^2 \):
\[ (x + iy)^2 = x^2 + 2xyi - y^2 = (x^2 - y^2) + (2xy)i \]
So, we can rewrite the equation as:
\[ (x^2 - y^2 + 2xyi) + (x - iy) = 0 \]

### Step 4: Combine real and imaginary parts
Combining the real and imaginary parts, we have:
\[ (x^2 - y^2 + x) + (2xy - y)i = 0 \]
For this equation to hold, both the real part and the imaginary part must be equal to zero:
1. \( x^2 - y^2 + x = 0 \) (Real part)
2. \( 2xy - y = 0 \) (Imaginary part)

### Step 5: Solve the imaginary part equation
From the imaginary part \( 2xy - y = 0 \), we can factor out \( y \):
\[ y(2x - 1) = 0 \]
This gives us two cases:
1. \( y = 0 \)
2. \( 2x - 1 = 0 \) which implies \( x = \frac{1}{2} \)

### Step 6: Solve for \( x \) when \( y = 0 \)
Substituting \( y = 0 \) into the real part equation:
\[ x^2 + x = 0 \]
Factoring gives:
\[ x(x + 1) = 0 \]
Thus, \( x = 0 \) or \( x = -1 \). This gives us two solutions:
1. \( z = 0 + 0i = 0 \)
2. \( z = -1 + 0i = -1 \)

### Step 7: Solve for \( y \) when \( x = \frac{1}{2} \)
Substituting \( x = \frac{1}{2} \) into the real part equation:
\[ \left(\frac{1}{2}\right)^2 - y^2 + \frac{1}{2} = 0 \]
This simplifies to:
\[ \frac{1}{4} - y^2 + \frac{1}{2} = 0 \]
Combining terms:
\[ \frac{3}{4} - y^2 = 0 \implies y^2 = \frac{3}{4} \implies y = \pm \frac{\sqrt{3}}{2} \]
Thus, we have two more solutions:
1. \( z = \frac{1}{2} + i\frac{\sqrt{3}}{2} \)
2. \( z = \frac{1}{2} - i\frac{\sqrt{3}}{2} \)

### Final Solutions
In total, we have found four solutions:
1. \( z = 0 \)
2. \( z = -1 \)
3. \( z = \frac{1}{2} + i\frac{\sqrt{3}}{2} \)
4. \( z = \frac{1}{2} - i\frac{\sqrt{3}}{2} \)

### Conclusion
The number of solutions to the equation \( z^2 + \overline{z} = 0 \) is **4**.

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