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`
B. `b=-c`
C. `c=-a`
D. `b=a+c`

Answer

To solve the problem, we need to find the relationship between the coefficients \( A \), \( B \), and \( C \) of the quadratic equation \( ax^2 + bx + c = 0 \) given that decreasing each root of this equation by 1 results in the equation \( 2x^2 + 8x + 2 = 0 \).

### Step-by-Step Solution:

1. **Identify the roots of the new equation**:
The equation \( 2x^2 + 8x + 2 = 0 \) can be expressed in terms of its roots. Let the roots of this equation be \( \alpha \) and \( \beta \). According to Vieta's formulas:
\[ \alpha + \beta = -\frac{b}{a} = -\frac{8}{2} = -4 \]
\[ \alpha \beta = \frac{c}{a} = \frac{2}{2} = 1 \]

2. **Relate the roots of the original equation**:
If the roots of the original equation \( ax^2 + bx + c = 0 \) are \( r_1 \) and \( r_2 \), then the roots of the new equation formed by decreasing each root by 1 are \( r_1 - 1 \) and \( r_2 - 1 \). Therefore:
\[ r_1 - 1 = \alpha \quad \text{and} \quad r_2 - 1 = \beta \]

3. **Express the original roots in terms of \( \alpha \) and \( \beta \)**:
From the above, we can express \( r_1 \) and \( r_2 \) as:
\[ r_1 = \alpha + 1 \quad \text{and} \quad r_2 = \beta + 1 \]

4. **Calculate the sum and product of the original roots**:
Using Vieta's formulas again for the original equation:
\[ r_1 + r_2 = -\frac{b}{a} \quad \text{and} \quad r_1 r_2 = \frac{c}{a} \]
Substituting for \( r_1 \) and \( r_2 \):
\[ (\alpha + 1) + (\beta + 1) = -\frac{b}{a} \]
Simplifying gives:
\[ \alpha + \beta + 2 = -\frac{b}{a} \]
Substituting \( \alpha + \beta = -4 \):
\[ -4 + 2 = -\frac{b}{a} \implies -2 = -\frac{b}{a} \implies \frac{b}{a} = 2 \implies b = 2a \]

5. **Calculate the product of the original roots**:
Now for the product:
\[ (\alpha + 1)(\beta + 1) = \frac{c}{a} \]
Expanding gives:
\[ \alpha \beta + \alpha + \beta + 1 = \frac{c}{a} \]
Substituting \( \alpha \beta = 1 \) and \( \alpha + \beta = -4 \):
\[ 1 - 4 + 1 = \frac{c}{a} \implies -2 = \frac{c}{a} \implies c = -2a \]

6. **Final Relationships**:
From our calculations, we have:
\[ b = 2a \quad \text{and} \quad c = -2a \]

### Conclusion:
The relationships between the coefficients are:
\[ B = 2A \quad \text{and} \quad C = -2A \]