Let `z_(1),z_(2) and z_(3)` be non-zero complex numbers satisfying `z^(2)=bar(iz), where i=sqrt(-1).`
What is `z_(1)+z_(2)+z_(3)` equal to ?

Options

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

Explanation

To solve the problem, we start with the equation given for the complex numbers \( z_1, z_2, z_3 \):

\[ z^2 = i \bar{z} \]

where \( i = \sqrt{-1} \).

### 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 of \( z \) is given by:

\[ \bar{z} = x - iy \]

### Step 2: Substitute \( z \) and \( \bar{z} \) into the equation
Substituting \( z \) and \( \bar{z} \) into the equation:

\[ (x + iy)^2 = i(x - iy) \]

### Step 3: Expand both sides
Expanding the left side:

\[ x^2 + 2xyi - y^2 = i(x - iy) \]

Expanding the right side:

\[ ix + y \]

### Step 4: Equate real and imaginary parts
Now, equate the real and imaginary parts from both sides:

1. Real part: \( x^2 - y^2 = y \)
2. Imaginary part: \( 2xy = x \)

### Step 5: Solve the imaginary part equation
From the imaginary part equation \( 2xy = x \), we can factor it as:

\[ x(2y - 1) = 0 \]

This gives us two cases:

1. \( x = 0 \)
2. \( 2y - 1 = 0 \) which implies \( y = \frac{1}{2} \)

### Step 6: Solve for \( y \) when \( x = 0 \)
If \( x = 0 \), substitute into the real part equation:

\[ 0 - y^2 = y \implies y^2 + y = 0 \implies y(y + 1) = 0 \]

This gives us:

1. \( y = 0 \)
2. \( y = -1 \)

So, we have two complex numbers:

- \( z_1 = 0 + 0i = 0 \) (not valid since we need non-zero)
- \( z_2 = 0 - i = -i \) (valid)

### Step 7: Solve for \( x \) when \( y = \frac{1}{2} \)
Now, substituting \( y = \frac{1}{2} \) into the real part equation:

\[ x^2 - \left(\frac{1}{2}\right)^2 = \frac{1}{2} \]

This simplifies to:

\[ x^2 - \frac{1}{4} = \frac{1}{2} \implies x^2 = \frac{1}{2} + \frac{1}{4} = \frac{3}{4} \]

Thus, we find:

\[ x = \pm \frac{\sqrt{3}}{2} \]

### Step 8: List the valid complex numbers
The valid complex numbers are:

1. \( z_2 = \frac{\sqrt{3}}{2} + i\frac{1}{2} \)
2. \( z_3 = -\frac{\sqrt{3}}{2} + i\frac{1}{2} \)

### Step 9: Calculate \( z_1 + z_2 + z_3 \)
Now, we can compute:

\[ z_1 + z_2 + z_3 = 0 + \left(\frac{\sqrt{3}}{2} + i\frac{1}{2}\right) + \left(-\frac{\sqrt{3}}{2} + i\frac{1}{2}\right) \]

This simplifies to:

\[ z_1 + z_2 + z_3 = 0 + 0 + i = i \]

### Conclusion
Thus, the sum \( z_1 + z_2 + z_3 \) is:

\[ \boxed{0} \]