For the complex number Z, the sum of all the solutions of `Z^(2)+|Z|=(barZ)^(2)` is equal to

Answer

To solve the equation \( Z^2 + |Z| = (\bar{Z})^2 \) for the complex number \( Z \), we will follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ Z^2 + |Z| = (\bar{Z})^2 \]
Recall that \( \bar{Z} \) is the conjugate of \( Z \). We can express \( Z \) in terms of its real and imaginary parts. Let \( Z = x + iy \), where \( x \) and \( y \) are real numbers. Then, \( |Z| = \sqrt{x^2 + y^2} \) and \( \bar{Z} = x - iy \).

### Step 2: Substitute the conjugate
Substituting \( \bar{Z} \) into the equation gives:
\[ Z^2 + |Z| = (x - iy)^2 \]
Calculating \( (x - iy)^2 \):
\[ (x - iy)^2 = x^2 - 2xyi - y^2 = x^2 - y^2 - 2xyi \]
Thus, the equation becomes:
\[ Z^2 + |Z| = x^2 - y^2 - 2xyi \]

### Step 3: Express \( Z^2 \)
Next, we express \( Z^2 \):
\[ Z^2 = (x + iy)^2 = x^2 + 2xyi - y^2 = x^2 - y^2 + 2xyi \]
Now, substituting back into the equation gives:
\[ (x^2 - y^2 + 2xyi) + |Z| = x^2 - y^2 - 2xyi \]

### Step 4: Separate real and imaginary parts
Now, we separate the real and imaginary parts:
- Real part: \( x^2 - y^2 + |Z| = x^2 - y^2 \)
- Imaginary part: \( 2xy + 0 = -2xy \)

From the real part, we can simplify:
\[ |Z| = 0 \]
From the imaginary part, we have:
\[ 2xy = -2xy \implies 4xy = 0 \]
This implies either \( x = 0 \) or \( y = 0 \).

### Step 5: Solve for \( Z \)
Given \( |Z| = 0 \), we conclude that:
\[ Z = 0 + 0i = 0 \]
Thus, the only solution is \( Z = 0 \).

### Step 6: Sum of all solutions
Since the only solution is \( Z = 0 \), the sum of all solutions is:
\[ \text{Sum} = 0 \]

### Final Answer
The sum of all the solutions of \( Z^2 + |Z| = (\bar{Z})^2 \) is equal to:
\[ \boxed{0} \]