Question

Let \( f(x) = px^2 + qx + r \), where \( p,q,r \) are constants and \( p \neq 0 \). If \( f(5) = -3f(2) \) and \( f(-4) = 0 \), then the other root of \( f \) is

• A. \( 3 \)
• B. \( -7 \)
• C. \( -2 \)
• D. \( 2 \)
• E. \( 6 \)

Hint:

Use given root condition to reduce unknowns in quadratic equations.

Answer 1

The Correct Option is A

Concept: If one root is known, use: \[ \alpha + \beta = -\frac{q}{p} \]

Step 1: Given \( f(-4)=0 \Rightarrow -4 \) is a root.

Step 2: Use condition: \[ f(5) = -3f(2) \] Substitute: \[ 25p +5q + r = -3(4p +2q + r) \]

Step 3: Simplify.
\[ 25p +5q + r = -12p -6q -3r \] \[ 37p +11q +4r = 0 \]

Step 4: Using root \( -4 \):
\[ 16p -4q + r = 0 \]

Step 5: Solve system → find ratio of coefficients → other root = 3.