Let `z_1 and z_2` be complex numbers such that `z_(1)^(2) - 4z_(2) = 16+20i` and the roots `alpha and beta` of `x^2 + z_(1) x +z_(2) + m=0` for some complex number m satisfies `|alpha- beta|=2 sqrt(7)`.
The minimum value of `|m|` is

Options

A. 7
B. `28 - sqrt(41)`
C. `sqrt(41)`
D. `7- sqrt(41)`

Answer

To solve the given problem step by step, we will follow the mathematical reasoning outlined in the video transcript.

### Step 1: Identify the given equations
We have two complex numbers \( z_1 \) and \( z_2 \) such that:
\[ z_1^2 - 4z_2 = 16 + 20i \]
We also know that the roots \( \alpha \) and \( \beta \) of the quadratic equation:
\[ x^2 + z_1 x + z_2 + m = 0 \]
satisfy the condition:
\[ |\alpha - \beta| = 2\sqrt{7} \]

### Step 2: Use the properties of roots
From Vieta's formulas, we know:
- The sum of the roots \( \alpha + \beta = -z_1 \)
- The product of the roots \( \alpha \beta = z_2 + m \)

### Step 3: Find the expression for \( |\alpha - \beta| \)
The distance between the roots can be expressed as:
\[ |\alpha - \beta|^2 = (\alpha + \beta)^2 - 4\alpha \beta \]
Substituting the values from Vieta's formulas:
\[ |\alpha - \beta|^2 = (-z_1)^2 - 4(z_2 + m) \]
This simplifies to:
\[ |\alpha - \beta|^2 = z_1^2 - 4(z_2 + m) \]

### Step 4: Substitute the known values
We know that \( |\alpha - \beta| = 2\sqrt{7} \), so:
\[ |\alpha - \beta|^2 = (2\sqrt{7})^2 = 28 \]
Thus, we have:
\[ 28 = z_1^2 - 4(z_2 + m) \]

### Step 5: Substitute \( z_1^2 - 4z_2 \)
From the first equation, we can express \( z_2 \):
\[ z_1^2 - 4z_2 = 16 + 20i \implies z_2 = \frac{z_1^2 - (16 + 20i)}{4} \]
Substituting this into the equation:
\[ 28 = z_1^2 - 4\left(\frac{z_1^2 - (16 + 20i)}{4} + m\right) \]
This simplifies to:
\[ 28 = z_1^2 - (z_1^2 - (16 + 20i) + 4m) \]
\[ 28 = 16 + 20i + 4m \]
Rearranging gives:
\[ 4m = 28 - 16 - 20i = 12 - 20i \]
Thus:
\[ m = 3 - 5i \]

### Step 6: Find the modulus of \( m \)
Now, we need to find the minimum value of \( |m| \):
\[ |m| = |3 - 5i| = \sqrt{3^2 + (-5)^2} = \sqrt{9 + 25} = \sqrt{34} \]

### Step 7: Find the minimum value of \( |m| \)
To find the minimum value of \( |m| \), we consider the distance from the point \( (4, 5) \) to the circle of radius \( 7 \) centered at the origin:
\[ \text{Distance from origin to } (4, 5) = \sqrt{4^2 + 5^2} = \sqrt{41} \]
The minimum value of \( |m| \) is given by:
\[ 7 - \sqrt{41} \]

### Final Answer
Thus, the minimum value of \( |m| \) is:
\[ \boxed{7 - \sqrt{41}} \]