The equation formed by decreasing each root of `ax^(2) + bx + c = 0` by 2 is `x^(2) + 4x +3= 0` then

Options

A. b = 1, a + c = 0
B. b = 2, a + c = 0
C. b = 0, a + c = 0
D. b = 0, a-c=0

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 by 2 leads to the equation \( x^2 + 4x + 3 = 0 \).

### Step-by-Step Solution:

1. **Identify the Roots of the Given Equation**:
The equation \( x^2 + 4x + 3 = 0 \) can be factored as:
\[ (x + 3)(x + 1) = 0 \]
Thus, the roots are:
\[ x = -3 \quad \text{and} \quad x = -1 \]

2. **Determine the Original Roots**:
According to the problem, these roots are obtained by decreasing the roots of the original equation \( ax^2 + bx + c = 0 \) by 2. Therefore, we can find the original roots by adding 2 to each of the roots:
\[ \text{Original root 1} = -3 + 2 = -1 \]
\[ \text{Original root 2} = -1 + 2 = 1 \]

3. **Form the Original Equation**:
The original roots are \( -1 \) and \( 1 \). The quadratic equation with these roots can be expressed as:
\[ (x + 1)(x - 1) = 0 \]
This simplifies to:
\[ x^2 - 1 = 0 \]

4. **Relate to the General Form**:
The general form of a quadratic equation is:
\[ ax^2 + bx + c = 0 \]
From the equation \( x^2 - 1 = 0 \), we can see that:
\[ a = 1, \quad b = 0, \quad c = -1 \]

5. **Establish Relationships**:
From the coefficients, we have:
\[ b = 0 \quad \text{and} \quad c = -a \]
This implies:
\[ a + c = 0 \]

### Final Result:
Thus, the relationships derived from the problem are:
\[ b = 0 \quad \text{and} \quad a + c = 0 \]