the equation formed by decreasing each root of `ax ^2 +bx+C=0` by 1 is ` 2x^2 + 8x +2=0` then

Options

A. a+b=0
B. b+c=0
C. b=c
D. a=b

Answer

To solve the problem, we need to find the values of \( a \), \( b \), and \( c \) based on the given quadratic equation formed by decreasing each root of \( ax^2 + bx + c = 0 \) by 1, which results in the equation \( 2x^2 + 8x + 2 = 0 \).

### Step-by-Step Solution:

1. **Identify the Given Equation**:
The given equation is:
\[ 2x^2 + 8x + 2 = 0 \]
We can simplify this by dividing through by 2:
\[ x^2 + 4x + 1 = 0 \]

2. **Let the Roots of the Original Equation be \( \alpha \) and \( \beta \)**:
The roots of the original equation \( ax^2 + bx + c = 0 \) are \( \alpha \) and \( \beta \).

3. **Relation of Roots**:
If we decrease each root by 1, the new roots become \( \alpha - 1 \) and \( \beta - 1 \). The equation with these new roots can be expressed as:
\[ (x - (\alpha - 1))(x - (\beta - 1)) = 0 \]
Expanding this gives:
\[ (x - \alpha + 1)(x - \beta + 1) = 0 \]
\[ = (x - \alpha)(x - \beta) + (x - \alpha) + (x - \beta) + 1 = 0 \]
\[ = x^2 - (\alpha + \beta)x + \alpha\beta + 1 \]

4. **Comparing Coefficients**:
From the equation \( x^2 + 4x + 1 = 0 \), we can compare coefficients:
- Coefficient of \( x \):
\[ -(\alpha + \beta) = 4 \implies \alpha + \beta = -4 \]
- Constant term:
\[ \alpha\beta + 1 = 1 \implies \alpha\beta = 0 \]

5. **Using Vieta's Formulas**:
From Vieta's formulas for the original equation \( ax^2 + bx + c = 0 \):
- \( \alpha + \beta = -\frac{b}{a} \)
- \( \alpha\beta = \frac{c}{a} \)

Substituting the known values:
\[ -\frac{b}{a} = -4 \implies b = 4a \]
\[ \frac{c}{a} = 0 \implies c = 0 \]

6. **Finding \( a \), \( b \), and \( c \)**:
Since \( c = 0 \), we can substitute this into the equation \( b = 4a \):
- Let \( a = 1 \) (for simplicity), then \( b = 4 \times 1 = 4 \).
- Thus, we have \( a = 1 \), \( b = 4 \), and \( c = 0 \).

7. **Final Values**:
Therefore, we have:
- \( a = 1 \)
- \( b = 4 \)
- \( c = 0 \)

### Conclusion:
The values of \( a \), \( b \), and \( c \) satisfy the relationships derived from the problem.