Question

If the equations $x^2 + px + 2 = 0$ and $x^2 + x + 2p = 0$ have a common root then the sum of the roots of the equation $x^2 + 2px + 8 = 0$ is

• A. -3
• B. 3
• C. 6
• D. -6

Hint:

When two polynomial equations $f(x)=0$ and $g(x)=0$ have a common root, that root is also a root of $f(x)-g(x)=0$. This often simplifies the problem by reducing the degree of the equation you need to solve. Always check all possible cases that arise from the factorization.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
If two quadratic equations have a common root, that root must satisfy both equations. We can solve the two equations to find a condition on the coefficients, and then find the value of the common root.
Step 2: Key Formula or Approach
Let $\alpha$ be the common root of the equations $x^2 + px + 2 = 0$ and $x^2 + x + 2p = 0$.
Then $\alpha$ must satisfy both:
1) $\alpha^2 + p\alpha + 2 = 0$
2) $\alpha^2 + \alpha + 2p = 0$
Subtracting equation (2) from equation (1) will eliminate the $\alpha^2$ term, allowing us to solve for $\alpha$.
Step 3: Detailed Explanation
Subtracting the two equations:
\[ (\alpha^2 + p\alpha + 2) - (\alpha^2 + \alpha + 2p) = 0 \] \[ p\alpha - \alpha + 2 - 2p = 0 \] \[ \alpha(p - 1) - 2(p - 1) = 0 \] \[ (\alpha - 2)(p - 1) = 0 \] This gives two possibilities: $p=1$ or $\alpha=2$.
Case 1: $p=1$.
If $p=1$, both equations become $x^2+x+2=0$. The discriminant of this equation is $D = b^2-4ac = 1^2 - 4(1)(2) = -7<0$. The roots are imaginary, but they are both common. So $p=1$ is a valid possibility.
Case 2: $\alpha=2$.
If the common root is $\alpha=2$, it must satisfy either equation. Let's substitute it into the first equation:
\[ (2)^2 + p(2) + 2 = 0 \] \[ 4 + 2p + 2 = 0 \] \[ 2p + 6 = 0 \implies p = -3 \]
Let's check if $\alpha=2$ also satisfies the second equation with $p=-3$:
\[ (2)^2 + (2) + 2(-3) = 4 + 2 - 6 = 0 \]
It does. So $p=-3$ is also a valid possibility.

The question doesn't specify any conditions on $p$ or the roots. Let's evaluate the required sum for both values of $p$.
The final equation is $x^2 + 2px + 8 = 0$.
The sum of the roots of a quadratic equation $ax^2+bx+c=0$ is $-b/a$.
Sum of roots = $-2p$.
If $p=1$, the sum of the roots is $-2(1) = -2$. This is not among the options.
If $p=-3$, the sum of the roots is $-2(-3) = 6$. This is option (C).
Step 4: Final Answer
We found two possible values for $p$: $p=1$ and $p=-3$.
For $p=1$, the sum of roots of $x^2 + 2px + 8 = 0$ is $-2$.
For $p=-3$, the sum of roots of $x^2 + 2px + 8 = 0$ is $6$.
Since $6$ is one of the options, we choose this as the intended answer.