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

Options

A. a+c
B. a-c
C. ac
D. `a//c`

Answer

To solve the problem, we start with the two quadratic equations given:

1. \( x^2 - cx + d = 0 \) (Equation 1)
2. \( x^2 - ax + b = 0 \) (Equation 2)

We know that these two equations have one common root, which we will denote as \( \alpha \), and that the second equation has equal roots.

### Step 1: Identify the roots of Equation 2
Since Equation 2 has equal roots, we can express its roots as \( \alpha \) and \( \alpha \). According to Vieta's formulas, the sum of the roots of the quadratic equation \( x^2 - ax + b = 0 \) is given by:

\[ \alpha + \alpha = a \implies 2\alpha = a \implies \alpha = \frac{a}{2} \]

### Step 2: Find the product of the roots of Equation 2
The product of the roots of Equation 2 is given by:

\[ \alpha \cdot \alpha = b \implies \alpha^2 = b \]

### Step 3: Substitute \( \alpha \) into Equation 1
Since \( \alpha \) is also a root of Equation 1, we can substitute \( \alpha \) into Equation 1:

\[ \alpha^2 - c\alpha + d = 0 \]

### Step 4: Replace \( \alpha^2 \) with \( b \) and \( \alpha \) with \( \frac{a}{2} \)
Substituting \( \alpha^2 = b \) and \( \alpha = \frac{a}{2} \) into the equation gives:

\[ b - c\left(\frac{a}{2}\right) + d = 0 \]

### Step 5: Rearranging the equation
Rearranging the equation, we have:

\[ b + d = c\left(\frac{a}{2}\right) \]

### Step 6: Multiply the entire equation by 2
To find \( 2(b + d) \), we multiply the equation by 2:

\[ 2(b + d) = ca \]

### Conclusion
Thus, we have:

\[ 2(b + d) = ca \]

This means that \( 2(b + d) \) is equal to \( ac \).

### Final Answer
So, the answer is \( 2(b + d) = ac \).

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