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: Write down the given equation
We start with the equation:
\[ z^2 + |z| = \bar{z}^2 \]

### Step 2: Use the property of conjugates
Recall that the conjugate of a complex number \( z \) can be expressed as \( \bar{z} = x - iy \) if \( z = x + iy \). The modulus \( |z| \) is given by \( |z| = \sqrt{x^2 + y^2} \).

### Step 3: Rewrite the conjugate squared
The conjugate squared can be expressed as:
\[ \bar{z}^2 = (x - iy)^2 = x^2 - 2xyi - y^2 = (x^2 - y^2) - 2xyi \]

### Step 4: Substitute \( \bar{z}^2 \) into the equation
Substituting \( \bar{z}^2 \) into the original equation gives:
\[ z^2 + |z| = (x^2 - y^2) - 2xyi \]

### Step 5: Take the conjugate of the entire equation
Taking the conjugate of the entire equation \( z^2 + |z| = \bar{z}^2 \) gives us:
\[ \bar{z}^2 + |z| = z^2 \]

### Step 6: Set up the second equation
This gives us our second equation:
\[ \bar{z}^2 + |z| = z^2 \]

### Step 7: Add the two equations
Now we have two equations:
1. \( z^2 + |z| = \bar{z}^2 \)
2. \( \bar{z}^2 + |z| = z^2 \)

Adding these two equations:
\[ (z^2 + |z|) + (\bar{z}^2 + |z|) = \bar{z}^2 + z^2 \]
This simplifies to:
\[ 2|z| = 0 \]

### Step 8: Solve for \( z \)
From \( 2|z| = 0 \), we conclude that:
\[ |z| = 0 \]
This implies that:
\[ z = 0 \]

### Step 9: Find the sum of all solutions
Since the only solution we found is \( z = 0 \), the sum of all solutions is:
\[ \text{Sum of all solutions} = 0 \]

### Final Answer
The sum of all the solutions of the given equation is:
\[ \boxed{0} \]