If z is a complex number satisfying `|z^(2)+1|=4|z|`, then the minimum value of `|z|` is

Options

A. `2sqrt(5)+4`
B. `2sqrt(5)-4`
C. `sqrt(5)-2`
D. none of these

Explanation

To solve the problem, we start with the given equation:

\[ |z^2 + 1| = 4|z| \]

### Step 1: Rewrite the equation
We can rewrite the equation as:

\[ \frac{|z^2 + 1|}{|z|} = 4 \]

### Step 2: Express in terms of \( |z| \)
Let \( r = |z| \). Then we have:

\[ |z^2 + 1| = |z|^2 + 1 = r^2 + 1 \]

Thus, we can rewrite our equation as:

\[ \frac{r^2 + 1}{r} = 4 \]

### Step 3: Clear the fraction
Multiplying both sides by \( r \) (assuming \( r \neq 0 \)) gives:

\[ r^2 + 1 = 4r \]

### Step 4: Rearrange the equation
Rearranging this gives us a standard quadratic equation:

\[ r^2 - 4r + 1 = 0 \]

### Step 5: Solve the quadratic equation
We can use the quadratic formula \( r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 1, b = -4, c = 1 \):

\[ r = \frac{4 \pm \sqrt{(-4)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} \]

Calculating the discriminant:

\[ r = \frac{4 \pm \sqrt{16 - 4}}{2} = \frac{4 \pm \sqrt{12}}{2} = \frac{4 \pm 2\sqrt{3}}{2} = 2 \pm \sqrt{3} \]

### Step 6: Determine the minimum value
The two possible values for \( r \) are:

\[ r = 2 + \sqrt{3} \quad \text{and} \quad r = 2 - \sqrt{3} \]

Since \( \sqrt{3} \approx 1.732 \), we find:

\[ 2 - \sqrt{3} \approx 0.268 \]

Thus, the minimum value of \( |z| \) is:

\[ \boxed{2 - \sqrt{3}} \]