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|=2sqrt(7)`. The value of `|m|`,

Options

A. `7`
B. `28-sqrt(41)`
C. `sqrt(41)`
D. `2sqrt(6)-4`

Explanation

To solve the problem step by step, we will analyze the given equations and conditions.

### Step 1: Understand the equations
We are given two equations involving complex numbers \( z_1 \) and \( z_2 \):
1. \( z_1^2 - 4z_2 = 16 + 20i \)
2. 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: Use properties of roots
From the properties of the roots of a quadratic equation, we know:
- The sum of the roots \( \alpha + \beta = -z_1 \)
- The product of the roots \( \alpha \beta = z_2 + m \)

### Step 3: Calculate \( |\alpha - \beta| \)
We can express \( |\alpha - \beta| \) in terms of \( z_1 \) and \( m \):
\[ |\alpha - \beta| = \sqrt{(\alpha + \beta)^2 - 4\alpha\beta} \]
Substituting the expressions for \( \alpha + \beta \) and \( \alpha \beta \):
\[ |\alpha - \beta| = \sqrt{(-z_1)^2 - 4(z_2 + m)} \]
Given that \( |\alpha - \beta| = 2\sqrt{7} \), we can square both sides:
\[ (2\sqrt{7})^2 = (-z_1)^2 - 4(z_2 + m) \]
This simplifies to:
\[ 28 = z_1^2 - 4(z_2 + m) \]

### Step 4: Substitute \( z_2 \)
From the first equation \( z_1^2 - 4z_2 = 16 + 20i \), we can express \( z_2 \):
\[ 4z_2 = z_1^2 - (16 + 20i) \implies z_2 = \frac{z_1^2 - (16 + 20i)}{4} \]

### Step 5: Substitute \( z_2 \) in the equation
Now substitute \( z_2 \) back into the equation derived from the roots:
\[ 28 = z_1^2 - 4\left(\frac{z_1^2 - (16 + 20i)}{4} + m\right) \]
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,
\[ m = \frac{12 - 20i}{4} = 3 - 5i \]

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

### Final Answer
The value of \( |m| \) is \( \sqrt{34} \).

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