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: Find Common Root Condition Let \( \alpha \) be the common root. 1) \( \alpha^2 + p\alpha + 2 = 0 \) 2) \( \alpha^2 + \alpha + 2p = 0 \) Subtract (2) from (1): \( (p-1)\alpha + (2-2p) = 0 \) \( (p-1)\alpha - 2(p-1) = 0 \) \( (p-1)(\alpha - 2) = 0 \) Two cases: Case 1: \( p = 1 \). Equations become identical (\( x^2+x+2=0 \)). Case 2: \( \alpha = 2 \).
Step 2: Determine p Substitute \( \alpha = 2 \) into Eq (1): \( 2^2 + p(2) + 2 = 0 \) \( 4 + 2p + 2 = 0 \implies 2p = -6 \implies p = -3 \).
Step 3: Analyze the Third Equation The target equation is \( x^2 + 2px + 8 = 0 \). If \( p = 1 \), equation is \( x^2 + 2x + 8 = 0 \), sum of roots = -2 (Not in options). If \( p = -3 \), equation is \( x^2 - 6x + 8 = 0 \). Sum of roots = \( -(\text{coefficient of } x) / (\text{coefficient of } x^2) = -(-6)/1 = 6 \).