Question

If \( x = 2 \) is a common root of both the equations \( 2x^2 + 2x + p = 0 \) and \( qx^2 + qx + 18 = 0 \), then the value of \( (q - p) \) is:

• A. -4
• B. -3
• C. 9
• D. 4

Hint:

For a common root, substitute the value of \( x \) into both equations and solve for the constants.

Answer 1

The Correct Option is B

Given that \( x = 2 \) is a common root for both quadratic equations, substitute \( x = 2 \) into both equations. **Substituting \( x = 2 \) into the first equation \( 2x^2 + 2x + p = 0 \):** \[ 2(2)^2 + 2(2) + p = 0 \quad \Rightarrow \quad 8 + 4 + p = 0 \quad \Rightarrow \quad p = -12. \] **Substituting \( x = 2 \) into the second equation \( qx^2 + qx + 18 = 0 \):** \[ q(2)^2 + q(2) + 18 = 0 \quad \Rightarrow \quad 4q + 2q + 18 = 0 \quad \Rightarrow \quad 6q + 18 = 0 \quad \Rightarrow \quad q = -3. \] Now, find \( q - p \): \[ q - p = -3 - (-12) = -3 + 12 = 9. \] Thus, \( q - p = \boxed{9} \).