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 problem step by step, we will follow the reasoning laid out in the video transcript while providing clear mathematical expressions.

### Step 1: Understand 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:
\[ |\alpha - \beta| = 2\sqrt{7} \]

### Step 2: Relate roots to coefficients
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: Express \( |\alpha - \beta| \)
The distance between the roots can be expressed as:
\[ |\alpha - \beta| = \sqrt{(\alpha + \beta)^2 - 4\alpha \beta} \]
Substituting the values from Vieta's formulas:
\[ |\alpha - \beta| = \sqrt{(-z_1)^2 - 4(z_2 + m)} \]
This simplifies to:
\[ |\alpha - \beta| = \sqrt{z_1^2 - 4z_2 - 4m} \]

### Step 4: Substitute the given condition
We know \( |\alpha - \beta| = 2\sqrt{7} \), so we square both sides:
\[ (2\sqrt{7})^2 = z_1^2 - 4z_2 - 4m \]
This gives:
\[ 28 = z_1^2 - 4z_2 - 4m \]
Rearranging this, we find:
\[ 4m = z_1^2 - 4z_2 - 28 \]
Thus,
\[ m = \frac{z_1^2 - 4z_2 - 28}{4} \]

### Step 5: Substitute \( z_1^2 - 4z_2 \)
From the first equation, we know:
\[ z_1^2 - 4z_2 = 16 + 20i \]
Substituting this into our equation for \( m \):
\[ m = \frac{(16 + 20i) - 28}{4} = \frac{-12 + 20i}{4} = -3 + 5i \]

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

### Step 7: Conclusion
The minimum value of \( |m| \) is:
\[ \boxed{\sqrt{34}} \]