Find the greatest value of the moduli of complex numbers z satisfying the equation `|z-(4)/(z)|=2`. What is the minimum value ?

Answer

To solve the problem of finding the greatest and minimum values of the moduli of complex numbers \( z \) satisfying the equation \( |z - \frac{4}{z}| = 2 \), we will follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ |z - \frac{4}{z}| = 2 \]

### Step 2: Use the property of moduli
Using the triangle inequality, we can express the left-hand side:
\[ |z - \frac{4}{z}| = |z| - |\frac{4}{z}| \]
Let \( |z| = t \). Then, we have:
\[ |z| = t \quad \text{and} \quad |\frac{4}{z}| = \frac{4}{|z|} = \frac{4}{t} \]
Thus, the equation becomes:
\[ |z| - |\frac{4}{z}| = t - \frac{4}{t} \]
So we rewrite our equation as:
\[ |t - \frac{4}{t}| = 2 \]

### Step 3: Set up the inequalities
This gives us two cases to consider:
1. \( t - \frac{4}{t} = 2 \)
2. \( t - \frac{4}{t} = -2 \)

### Step 4: Solve the first case
For the first case:
\[ t - \frac{4}{t} = 2 \]
Multiplying through by \( t \) (assuming \( t \neq 0 \)):
\[ t^2 - 4 = 2t \]
Rearranging gives:
\[ t^2 - 2t - 4 = 0 \]
Using the quadratic formula \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ t = \frac{2 \pm \sqrt{(-2)^2 - 4 \cdot 1 \cdot (-4)}}{2 \cdot 1} = \frac{2 \pm \sqrt{4 + 16}}{2} = \frac{2 \pm \sqrt{20}}{2} = \frac{2 \pm 2\sqrt{5}}{2} = 1 \pm \sqrt{5} \]

### Step 5: Solve the second case
For the second case:
\[ t - \frac{4}{t} = -2 \]
Multiplying through by \( t \):
\[ t^2 + 2t - 4 = 0 \]
Using the quadratic formula again:
\[ t = \frac{-2 \pm \sqrt{(2)^2 - 4 \cdot 1 \cdot (-4)}}{2 \cdot 1} = \frac{-2 \pm \sqrt{4 + 16}}{2} = \frac{-2 \pm \sqrt{20}}{2} = \frac{-2 \pm 2\sqrt{5}}{2} = -1 \pm \sqrt{5} \]

### Step 6: Collect all possible values
From the two cases, we have the following possible values for \( t \):
1. From the first case: \( t = 1 + \sqrt{5} \) and \( t = 1 - \sqrt{5} \)
2. From the second case: \( t = -1 + \sqrt{5} \) and \( t = -1 - \sqrt{5} \)

### Step 7: Determine valid values
Since \( t \) represents the modulus \( |z| \), it must be non-negative:
- \( 1 + \sqrt{5} \) is valid.
- \( 1 - \sqrt{5} \) is negative and thus invalid.
- \( -1 + \sqrt{5} \) is valid since \( \sqrt{5} \approx 2.236 \) gives \( -1 + 2.236 \approx 1.236 \).
- \( -1 - \sqrt{5} \) is negative and thus invalid.

### Step 8: Conclusion
The valid values for \( |z| \) are:
- Maximum: \( 1 + \sqrt{5} \)
- Minimum: \( -1 + \sqrt{5} \)

### Final Answer:
- Greatest value of \( |z| \) is \( 1 + \sqrt{5} \).
- Minimum value of \( |z| \) is \( -1 + \sqrt{5} \).