If the quadratic equations ` x^(2) +pqx +r=0 and z^(2) +prx +q=0` have a common root then the equation containing their other roots is/are

Options

A. `x^(2)-p(q+r)x + p^(2)qr=0`
B. `x^(2)p(q+r)+(q +r) x-pqr=0`
C. `p(q+r)x^(2) - (q+r)x+pqr=0`
D. `x^(2) +p(q+r)x-p^(2)qr=0`

Answer

To solve the problem, we need to find the equation containing the other roots of the two given quadratic equations that have a common root. Let's denote the common root as \( \alpha \).

### Step 1: Set up the equations
We have two quadratic equations:
1. \( x^2 + pqx + r = 0 \) (let's call this equation \( f(x) \))
2. \( z^2 + prx + q = 0 \) (let's call this equation \( g(z) \))

Since both equations have a common root \( \alpha \), we can write:
\[ f(\alpha) = 0 \]
\[ g(\alpha) = 0 \]

### Step 2: Substitute the common root into both equations
Substituting \( \alpha \) into the first equation:
\[ \alpha^2 + pq\alpha + r = 0 \quad \text{(1)} \]

Substituting \( \alpha \) into the second equation:
\[ \alpha^2 + pr\alpha + q = 0 \quad \text{(2)} \]

### Step 3: Equate the two equations
Since both equations equal zero, we can set them equal to each other:
\[ \alpha^2 + pq\alpha + r = \alpha^2 + pr\alpha + q \]

### Step 4: Simplify the equation
Cancel \( \alpha^2 \) from both sides:
\[ pq\alpha + r = pr\alpha + q \]

Rearranging gives:
\[ pq\alpha - pr\alpha + r - q = 0 \]
\[ (pq - pr)\alpha + (r - q) = 0 \]

### Step 5: Factor out common terms
Factoring out \( \alpha \):
\[ \alpha(pq - pr) = q - r \]

### Step 6: Solve for \( \alpha \)
From the equation above, we can express \( \alpha \):
\[ \alpha = \frac{q - r}{pq - pr} \quad \text{(if } pq \neq pr\text{)} \]

### Step 7: Find the other roots
Let the other root of the first equation be \( \beta \) and the other root of the second equation be \( \gamma \). We know:
- For the first equation \( x^2 + pqx + r = 0 \):
- The sum of the roots \( \alpha + \beta = -pq \)
- The product of the roots \( \alpha \beta = r \)

From the product of roots:
\[ \beta = \frac{r}{\alpha} \]

- For the second equation \( z^2 + prx + q = 0 \):
- The sum of the roots \( \alpha + \gamma = -pr \)
- The product of the roots \( \alpha \gamma = q \)

From the product of roots:
\[ \gamma = \frac{q}{\alpha} \]

### Step 8: Form the new quadratic equation
Now we can form a new quadratic equation with roots \( \beta \) and \( \gamma \):
The sum of the roots \( \beta + \gamma = \frac{r}{\alpha} + \frac{q}{\alpha} = \frac{r + q}{\alpha} \)

The product of the roots \( \beta \gamma = \left(\frac{r}{\alpha}\right)\left(\frac{q}{\alpha}\right) = \frac{rq}{\alpha^2} \)

Using the standard form of a quadratic equation \( x^2 - (\text{sum of roots})x + (\text{product of roots}) = 0 \):
\[ x^2 - \left(\frac{r + q}{\alpha}\right)x + \frac{rq}{\alpha^2} = 0 \]

### Final Step: Clear the denominators
Multiply through by \( \alpha^2 \) to eliminate the fractions:
\[ \alpha^2 x^2 - (r + q) \alpha x + rq = 0 \]

This is the equation containing the other roots.

### Conclusion
The equation containing the other roots is:
\[ \alpha^2 x^2 - (r + q) \alpha x + rq = 0 \]