Suppose `p ne q` and difference between the roots of `x^(2) + 2px + q = 0` is twice the difference between the roots of `x^(2) + qx + (1)/(4) p = 0` then |p + q + 4| is equal to _______

Answer

To solve the problem step by step, we will analyze the two quadratic equations provided and derive the necessary relationships.

### Step 1: Identify the first quadratic equation
The first quadratic equation is given as:
\[ x^2 + 2px + q = 0 \]

### Step 2: Calculate the difference between the roots of the first quadratic equation
For a quadratic equation \( ax^2 + bx + c = 0 \), the difference between the roots can be calculated using the formula:
\[ \text{Difference} = \sqrt{(sum \ of \ roots)^2 - 4 \times (product \ of \ roots)} \]

Here, the sum of the roots \( \alpha + \beta = -\frac{b}{a} = -\frac{2p}{1} = -2p \) and the product of the roots \( \alpha \beta = \frac{c}{a} = \frac{q}{1} = q \).

Thus, the difference between the roots is:
\[ \text{Difference}_1 = \sqrt{(-2p)^2 - 4q} = \sqrt{4p^2 - 4q} = 2\sqrt{p^2 - q} \]

### Step 3: Identify the second quadratic equation
The second quadratic equation is given as:
\[ x^2 + qx + \frac{1}{4}p = 0 \]

### Step 4: Calculate the difference between the roots of the second quadratic equation
For this equation, the sum of the roots \( \alpha_1 + \beta_1 = -\frac{q}{1} = -q \) and the product of the roots \( \alpha_1 \beta_1 = \frac{\frac{1}{4}p}{1} = \frac{1}{4}p \).

Thus, the difference between the roots is:
\[ \text{Difference}_2 = \sqrt{(-q)^2 - 4 \cdot \frac{1}{4}p} = \sqrt{q^2 - p} \]

### Step 5: Set up the equation based on the problem statement
According to the problem, the difference between the roots of the first quadratic equation is twice the difference between the roots of the second quadratic equation:
\[ 2\sqrt{p^2 - q} = 2 \cdot \sqrt{q^2 - p} \]

### Step 6: Simplify the equation
Dividing both sides by 2 gives:
\[ \sqrt{p^2 - q} = \sqrt{q^2 - p} \]

### Step 7: Square both sides
Squaring both sides results in:
\[ p^2 - q = q^2 - p \]

### Step 8: Rearranging the equation
Rearranging gives:
\[ p^2 - q^2 = q - p \]
This can be factored as:
\[ (p - q)(p + q) = -(p - q) \]

### Step 9: Factor out \( (p - q) \)
Since \( p \neq q \), we can divide both sides by \( (p - q) \):
\[ p + q = -1 \]

### Step 10: Calculate \( |p + q + 4| \)
Now we need to find \( |p + q + 4| \):
\[ |p + q + 4| = |-1 + 4| = |3| = 3 \]

### Final Answer
Thus, the value of \( |p + q + 4| \) is:
\[ \boxed{3} \]