If `(x-2)` is a common factor of the expressions `x^(2)+ax+b` and `x^(2)+cx+d`, then `(b-d)/(c-a)` is equal to

Options

A. 1
B. 2
C. -1
D. -2

Answer

To solve the problem, we need to find the value of \((b - d)/(c - a)\) given that \(x - 2\) is a common factor of the expressions \(x^2 + ax + b\) and \(x^2 + cx + d\).

### Step-by-Step Solution:

1. **Identify the common root**: Since \(x - 2\) is a common factor, we can substitute \(x = 2\) into both expressions.

2. **Substituting into the first expression**:
\[ x^2 + ax + b = 0 \quad \text{at } x = 2 \]
Substituting \(x = 2\):
\[ 2^2 + 2a + b = 0 \implies 4 + 2a + b = 0 \implies 2a + b = -4 \quad \text{(Equation 1)} \]

3. **Substituting into the second expression**:
\[ x^2 + cx + d = 0 \quad \text{at } x = 2 \]
Substituting \(x = 2\):
\[ 2^2 + 2c + d = 0 \implies 4 + 2c + d = 0 \implies 2c + d = -4 \quad \text{(Equation 2)} \]

4. **Setting the equations equal**: Since both expressions equal zero at \(x = 2\), we can set the results of Equation 1 and Equation 2 equal to each other:
\[ 2a + b = 2c + d \]

5. **Rearranging the equation**:
\[ b - d = 2c - 2a \]

6. **Dividing both sides by \(c - a\)**:
\[ \frac{b - d}{c - a} = \frac{2c - 2a}{c - a} \]

7. **Simplifying the right side**:
\[ \frac{b - d}{c - a} = 2 \cdot \frac{c - a}{c - a} = 2 \]

Thus, we conclude that:
\[ \frac{b - d}{c - a} = 2 \]

### Final Answer:
\[ \frac{b - d}{c - a} = 2 \]