Question

The value of $a+b$ if the equation $x^2+ax+b=0$ and $x^2+bx+a=0$ have a common root is

• A. -1.0
• B. -1.00
• C. -1

Answer 1

Answer: (A), (B), (C)

Let $\alpha$ be a common root of the equations.
Then $\alpha$ will satisfy the equations.
Thus,
${\alpha }^2+a.\alpha +b=0\cdots \left(i\right)$
${\alpha }^2+b.\alpha +a=0\cdots \left(ii\right)$
Subtracting the two equations, we get:
$\alpha (a-b)+(b-a)=0$
$\Rightarrow \alpha (a-b)-(a-b)=0$
$\Rightarrow \alpha (a-b)=(a-b)$
Since $a\ne b$
$\therefore \alpha =1$
Hence, placing the value of $\alpha$ in the equation (i), we get
$1^2+a.1+b=0\Rightarrow a+b=-1$
Hence, value of $a+b=-1$