If a root of the equations `x^(2)+px+q=0 and x^(2)+alpha x+beta=0` is common, then its value will be `("where "p!=alpha and q ne beta)`

Options

A. `(q-B)/(alpha-p)`
B. `(p beta-alpha q)/(q-B)`
C. `(q-B)/(alpha-p) or (p beta-alpha q)/(q-B)`
D. None of these

Answer

To find the common root of the equations \(x^2 + px + q = 0\) and \(x^2 + \alpha x + \beta = 0\), we will follow these steps:

### Step 1: Assume the common root
Let the common root be \(m\). This means that \(m\) satisfies both equations.

### Step 2: Write the equations for the common root
From the first equation \(x^2 + px + q = 0\), substituting \(m\) gives:
\[ m^2 + pm + q = 0 \quad \text{(1)} \]
From the second equation \(x^2 + \alpha x + \beta = 0\), substituting \(m\) gives:
\[ m^2 + \alpha m + \beta = 0 \quad \text{(2)} \]

### Step 3: Set the equations equal to each other
Since both equations equal zero, we can set them equal to each other:
\[ m^2 + pm + q = m^2 + \alpha m + \beta \]

### Step 4: Simplify the equation
Cancelling \(m^2\) from both sides, we have:
\[ pm + q = \alpha m + \beta \]
Rearranging gives:
\[ pm - \alpha m + q - \beta = 0 \]
Factoring out \(m\) gives:
\[ m(p - \alpha) + (q - \beta) = 0 \]

### Step 5: Solve for \(m\)
From the equation \(m(p - \alpha) + (q - \beta) = 0\), we can isolate \(m\):
\[ m(p - \alpha) = - (q - \beta) \]
Thus,
\[ m = \frac{-(q - \beta)}{(p - \alpha)} \quad \text{(if } p \neq \alpha\text{)} \]
This can also be written as:
\[ m = \frac{\beta - q}{\alpha - p} \]

### Step 6: Conclusion
The value of the common root \(m\) is:
\[ m = \frac{\beta - q}{\alpha - p} \]