If the two equations `x^2 - cx + d = 0` and `x^2- ax + b = 0` have one common root and the second equation has equal roots, then 2 (b + d) =

Answer

To solve the problem, we need to analyze the two given equations and use the properties of their roots.

### Step-by-Step Solution:

1. **Identify the Equations and Roots**:
We have two equations:
- \( x^2 - cx + d = 0 \) (let's denote the roots as \( \alpha \) and \( \beta \))
- \( x^2 - ax + b = 0 \) (this equation has equal roots, so both roots are \( \alpha \))

2. **Use the Properties of Roots**:
For the first equation:
- The sum of the roots \( \alpha + \beta = c \)
- The product of the roots \( \alpha \beta = d \)

For the second equation (having equal roots):
- The sum of the roots \( \alpha + \alpha = 2\alpha = a \)
- The product of the roots \( \alpha \cdot \alpha = \alpha^2 = b \)

3. **Express \( \beta \) in terms of \( \alpha \)**:
From the sum of the roots of the first equation, we can express \( \beta \):
\[ \beta = c - \alpha \]

4. **Substitute \( \beta \) into the Product of Roots**:
Using the product of the roots from the first equation:
\[ \alpha \beta = d \implies \alpha (c - \alpha) = d \]
Expanding this gives:
\[ \alpha c - \alpha^2 = d \quad \text{(1)} \]

5. **Substitute \( b \) from the Second Equation**:
From the second equation, we have:
\[ b = \alpha^2 \quad \text{(2)} \]

6. **Find \( 2(b + d) \)**:
We need to find \( 2(b + d) \):
\[ 2(b + d) = 2(\alpha^2 + d) \]
Substitute \( d \) from equation (1):
\[ 2(b + d) = 2(\alpha^2 + \alpha c - \alpha^2) = 2(\alpha c) \]

7. **Express \( \alpha \) in terms of \( a \)**:
From the sum of the roots of the second equation, we have:
\[ 2\alpha = a \implies \alpha = \frac{a}{2} \]

8. **Substitute \( \alpha \) into \( 2(b + d) \)**:
Now substituting \( \alpha = \frac{a}{2} \):
\[ 2(b + d) = 2\left(\frac{a}{2}c\right) = ac \]

### Final Result:
Thus, we conclude that:
\[ 2(b + d) = ac \]