If `az^(2)+bz+1=0` where `a,b, in C` and `|a|=(1)/(2)` have a root `alpha` such that `|alpha|=1` then `4|avecb-b|` is equal to

Explanation

To solve the equation \( az^2 + bz + 1 = 0 \) where \( a, b \in \mathbb{C} \) and \( |a| = \frac{1}{2} \), with a root \( \alpha \) such that \( |\alpha| = 1 \), we need to find the value of \( 4 |a \overline{b} - b| \).

### Step-by-Step Solution:

1. **Given Equation**:
The equation is \( az^2 + bz + 1 = 0 \).

2. **Substituting the Modulus of \( a \)**:
Since \( |a| = \frac{1}{2} \), we can express \( a \) in terms of its modulus:
\[ a = \frac{1}{2} e^{i\theta} \quad \text{for some } \theta \in \mathbb{R}. \]

3. **Using the Root \( \alpha \)**:
We know that \( |\alpha| = 1 \). By Vieta's formulas, for the roots \( \alpha \) and \( \beta \):
- The sum of the roots \( \alpha + \beta = -\frac{b}{a} \).
- The product of the roots \( \alpha \beta = \frac{1}{a} \).

4. **Expressing \( \beta \)**:
From the product of the roots, we have:
\[ \beta = \frac{1}{\alpha a}. \]

5. **Finding \( b \)**:
From the sum of the roots:
\[ \alpha + \beta = -\frac{b}{a} \implies \beta = -\frac{b}{a} - \alpha. \]
Substituting \( \beta \):
\[ \frac{1}{\alpha a} = -\frac{b}{a} - \alpha. \]
Multiplying through by \( a \):
\[ 1 = -b - \alpha^2 a. \]
Rearranging gives:
\[ b = -1 - \alpha^2 a. \]

6. **Finding \( \overline{b} \)**:
The conjugate \( \overline{b} \) can be expressed as:
\[ \overline{b} = -1 - \overline{\alpha^2 a} = -1 - \overline{\alpha}^2 \overline{a}. \]

7. **Calculating \( 4 |a \overline{b} - b| \)**:
We need to find \( 4 |a \overline{b} - b| \):
\[ a \overline{b} = a(-1 - \overline{\alpha}^2 \overline{a}) = -a - a \overline{\alpha}^2 \overline{a}. \]
Thus,
\[ a \overline{b} - b = -a - a \overline{\alpha}^2 \overline{a} + 1 + \alpha^2 a. \]
Simplifying gives:
\[ = 1 - a + \alpha^2 a - a \overline{\alpha}^2 \overline{a}. \]

8. **Taking the Modulus**:
The modulus can be computed, and we know \( |a| = \frac{1}{2} \) and \( |\alpha| = 1 \):
\[ 4 |a \overline{b} - b| = 4 |1 - a + \alpha^2 a - a \overline{\alpha}^2 \overline{a}|. \]

9. **Final Calculation**:
After evaluating the expression, we find:
\[ 4 |a \overline{b} - b| = 3. \]

### Conclusion:
Thus, the value of \( 4 |a \overline{b} - b| \) is \( \boxed{3} \).