Solve `|z|+z= 2+ i`, where z is a complex number

Answer

To solve the equation \( |z| + z = 2 + i \), where \( z \) is a complex number, we will proceed step by step.

### Step 1: Represent \( z \) in terms of real and imaginary parts
Let \( z = x + iy \), where \( x \) and \( y \) are real numbers. The modulus of \( z \) is given by:
\[ |z| = \sqrt{x^2 + y^2} \]

### Step 2: Substitute \( z \) and \( |z| \) into the equation
Substituting the expressions for \( z \) and \( |z| \) into the equation gives:
\[ \sqrt{x^2 + y^2} + (x + iy) = 2 + i \]

### Step 3: Separate real and imaginary parts
This can be rewritten as:
\[ \sqrt{x^2 + y^2} + x + iy = 2 + i \]
From this equation, we can separate the real and imaginary parts:
- Real part: \( \sqrt{x^2 + y^2} + x = 2 \)
- Imaginary part: \( y = 1 \)

### Step 4: Substitute \( y \) into the real part equation
Now, we substitute \( y = 1 \) into the real part equation:
\[ \sqrt{x^2 + 1^2} + x = 2 \]
This simplifies to:
\[ \sqrt{x^2 + 1} + x = 2 \]

### Step 5: Isolate the square root
Next, we isolate the square root:
\[ \sqrt{x^2 + 1} = 2 - x \]

### Step 6: Square both sides
Now we square both sides to eliminate the square root:
\[ x^2 + 1 = (2 - x)^2 \]
Expanding the right side:
\[ x^2 + 1 = 4 - 4x + x^2 \]

### Step 7: Simplify the equation
Subtract \( x^2 \) from both sides:
\[ 1 = 4 - 4x \]
Rearranging gives:
\[ 4x = 4 - 1 \]
\[ 4x = 3 \]
Thus,
\[ x = \frac{3}{4} \]

### Step 8: Write the solution for \( z \)
Now that we have both \( x \) and \( y \):
\[ z = x + iy = \frac{3}{4} + i \cdot 1 = \frac{3}{4} + i \]

### Final Answer
The solution to the equation \( |z| + z = 2 + i \) is:
\[ z = \frac{3}{4} + i \]