Question

If the equations x² + bx + c = 0, x² + cx + b = 0 [b ≠ c] have a common root, then ……………

A) b + c = 0 

B) b + c = 1 

C) b + c + 1 = 0 

D) None of these

Answer 1

Correct option is (C) b + c + 1 = 0

Let \(\alpha\) be common root of both given equations.

\(\therefore\) \(\alpha^2+b\alpha+c=0\)      ____________(1)

and \(\alpha^2+cx+b=0\)   ____________(2)

Subtract equation (1) from (2), we get

\((\alpha^2+c\alpha+b)-(\alpha^2+b\alpha+c)=0\)

\(\Rightarrow\) \((c-b)\alpha+b-c=0\)

\(\Rightarrow\) \((c-b)(\alpha-1)=0\)

\(\Rightarrow\) \(\alpha-1=0\)     \((\because b\neq c\Rightarrow b-c\neq0\Rightarrow c-b\neq0)\)

\(\Rightarrow\) \(\alpha\) = 1

Put \(\alpha\) = 1 in equation (1), we obtain

1+b+c = 0

Answer 2

Correct option is C) b + c + 1 = 0