Question

Two distinct polynomials $f\left(x\right)$ and $g\left(x\right)$ are defined as follows:
$f\left(x\right)=x^2+ax+2;g\left(x\right)=x^2+2x+a.$
If the equations $f\left(x\right)=0$ and $g\left(x\right)=0$ have a common root then the sum of the roots of the equation $f\left(x\right)+g\left(x\right)=0$ is

• A. $\frac{-1}{2}$
• B. $0$
• C. $\frac{1}{2}$
• D. $1$

Answer 1

The correct option is C $\frac{1}{2}$

Let $\alpha$ be the common root.
$\text{Then,}{\alpha }^2+a\alpha +2={\alpha }^2+2\alpha +a=0$
$\Rightarrow 2\alpha -a\alpha =2-a$
$\Rightarrow \alpha (2-a)=2-a$
$\Rightarrow \alpha =1$
Therefore, the other root of $f\left(x\right)=0$ is $2.$
$\Rightarrow f\left(x\right)=(x-1)(x-2)=x^2-3x+2$
$\therefore a=-3$
$\Rightarrow g\left(x\right)=x^2+2x-3$

$\text{Now,}f\left(x\right)+g\left(x\right)=0$
$\Rightarrow 2x^2-x-1=0$
Sum of the roots $=\frac{1}{2}$