Question

If \( x^2 + ax + b = 0 \) and \( x^2 + bx + a = 0 \) have exactly one common root, then the value of \( (a + b) \) is:

• A. 1
• B. 0
• C. -1
• D. 3

Hint:

When solving problems with common roots, subtract the equations and simplify to find relationships between the coefficients.

Answer 1

The Correct Option is B

We are given two quadratic equations: 1. \( x^2 + ax + b = 0 \) 2. \( x^2 + bx + a = 0 \) Let the common root be \( \alpha \). Therefore, we know: \[ \alpha^2 + a\alpha + b = 0 \quad \text{(Equation 1)} \] \[ \alpha^2 + b\alpha + a = 0 \quad \text{(Equation 2)} \] Subtract Equation 2 from Equation 1: \[ (\alpha^2 + a\alpha + b) - (\alpha^2 + b\alpha + a) = 0 \] \[ a\alpha + b - b\alpha - a = 0 \] \[ (a - b)\alpha = a - b \] If \( a \neq b \), we can divide both sides by \( a - b \), giving us: \[ \alpha = 1 \] Now, substitute \( \alpha = 1 \) into either Equation 1 or Equation 2. Using Equation 1: \[ 1^2 + a(1) + b = 0 \] \[ 1 + a + b = 0 \] \[ a + b = -1 \] Thus, the value of \( a + b \) is \( \boxed{0} \).