The equation `ax^(2)+bx+c=0 and x^(3)-4x^(2)+8x-8=0` have two roots in common. Then 2b + c is equal to _______.

Answer

To solve the problem, we need to find the value of \( 2b + c \) given that the quadratic equation \( ax^2 + bx + c = 0 \) and the cubic equation \( x^3 - 4x^2 + 8x - 8 = 0 \) have two roots in common.

### Step-by-Step Solution:

1. **Identify the Roots of the Cubic Equation:**
We start with the cubic equation:
\[ x^3 - 4x^2 + 8x - 8 = 0 \]
We need to find its roots. We can use the Rational Root Theorem or simply test some values.

2. **Testing Possible Roots:**
- Let's test \( x = 1 \):
\[ 1^3 - 4(1^2) + 8(1) - 8 = 1 - 4 + 8 - 8 = -3 \quad (\text{not a root}) \]
- Now, let's test \( x = 2 \):
\[ 2^3 - 4(2^2) + 8(2) - 8 = 8 - 16 + 16 - 8 = 0 \quad (\text{is a root}) \]

3. **Factor the Cubic Equation:**
Since \( x = 2 \) is a root, we can factor the cubic equation by dividing it by \( x - 2 \):
\[ \text{Using synthetic division or polynomial long division, we find:} \]
\[ x^3 - 4x^2 + 8x - 8 = (x - 2)(x^2 - 2x + 4) \]

4. **Finding the Roots of the Quadratic Factor:**
Now we need to find the roots of the quadratic \( x^2 - 2x + 4 = 0 \):
- Calculate the discriminant:
\[ D = b^2 - 4ac = (-2)^2 - 4(1)(4) = 4 - 16 = -12 \]
Since the discriminant is negative, the roots of this quadratic are non-real. We denote these roots as \( \alpha \) and \( \beta \).

5. **Identifying the Quadratic Equation:**
The quadratic equation \( ax^2 + bx + c = 0 \) has the same roots \( \alpha \) and \( \beta \). Therefore, the coefficients \( a, b, c \) must be proportional to those of \( x^2 - 2x + 4 \). We can set:
\[ a = 1, \quad b = -2, \quad c = 4 \]

6. **Calculating \( 2b + c \):**
Now we can calculate \( 2b + c \):
\[ 2b + c = 2(-2) + 4 = -4 + 4 = 0 \]

### Final Answer:
Thus, the value of \( 2b + c \) is:
\[ \boxed{0} \]