The number of complex numbers z which satisfy `z^(2) + 2|z|^(2) = 2` is

Options

A. 0
B. 2
C. 3
D. 4

Answer

To solve the problem of finding the number of complex numbers \( z \) that satisfy the equation

\[ z^2 + 2|z|^2 = 2, \]

we will follow these steps:

### Step 1: Express \( z \) in terms of its real and imaginary parts
Let \( z = x + iy \), where \( x \) is the real part and \( y \) is the imaginary part of the complex number \( z \).

### Step 2: Calculate \( |z|^2 \)
The modulus squared of \( z \) is given by:

\[ |z|^2 = x^2 + y^2. \]

### Step 3: Substitute \( z \) and \( |z|^2 \) into the equation
Substituting \( z \) and \( |z|^2 \) into the original equation, we have:

\[ (x + iy)^2 + 2(x^2 + y^2) = 2. \]

### Step 4: Expand \( (x + iy)^2 \)
Expanding \( (x + iy)^2 \):

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

### Step 5: Substitute the expansion back into the equation
Now substituting back, we get:

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

This simplifies to:

\[ (3x^2 - y^2) + 2xyi = 2. \]

### Step 6: Set the real and imaginary parts equal
For the equation to hold, both the real and imaginary parts must be equal to their corresponding parts on the right-hand side:

1. Real part: \( 3x^2 - y^2 = 2 \)
2. Imaginary part: \( 2xy = 0 \)

### Step 7: Solve the imaginary part equation
From \( 2xy = 0 \), we have two cases:

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

### Step 8: Case 1: \( x = 0 \)
Substituting \( x = 0 \) into the real part equation:

\[ 3(0)^2 - y^2 = 2 \implies -y^2 = 2 \implies y^2 = -2. \]

This has no real solutions.

### Step 9: Case 2: \( y = 0 \)
Substituting \( y = 0 \) into the real part equation:

\[ 3x^2 - (0)^2 = 2 \implies 3x^2 = 2 \implies x^2 = \frac{2}{3} \implies x = \pm \sqrt{\frac{2}{3}}. \]

### Step 10: Collect solutions
From \( y = 0 \), we have two solutions for \( x \):

1. \( \left(\sqrt{\frac{2}{3}}, 0\right) \)
2. \( \left(-\sqrt{\frac{2}{3}}, 0\right) \)

### Step 11: Combine solutions
Now, we combine the solutions from both cases:

- From \( x = 0 \): No solutions.
- From \( y = 0 \): Two solutions \( \left(\sqrt{\frac{2}{3}}, 0\right) \) and \( \left(-\sqrt{\frac{2}{3}}, 0\right) \).

### Step 12: Conclusion
Thus, the total number of complex numbers \( z \) that satisfy the equation is **2**.