Question

If the equations \( x^2 + ax + 1 = 0 \) and \( x^2 - x - a = 0 \) have a real common root, then the value of \( b \) is

• A. \( 0 \)
• B. \( 1 \)
• C. \( -1 \)
• D. \( 2 \)
• E. \( 3 \)

Hint:

Subtract equations to eliminate variables in common-root problems.

Answer 1

The Correct Option is C

Concept: Common root ⇒ both equations satisfied by same \( x \).

Step 1: Let common root be \( r \).
Then: \[ r^2 + ar + 1 = 0 \quad (1) \] \[ r^2 - r - a = 0 \quad (2) \]

Step 2: Subtract equations.
\[ (r^2 + ar + 1) - (r^2 - r - a) = 0 \] \[ ar + 1 + r + a = 0 \Rightarrow r(a+1) + (a+1) = 0 \]

Step 3: Factor.
\[ (a+1)(r+1) = 0 \]

Step 4: Possible cases.
Either: \[ a = -1 \quad \text{or} \quad r = -1 \]

Step 5: Substitute back to verify.
Valid condition gives: \[ a = -1 \] Thus required value: \[ -1 \]