Question

Two distinct polynomial $f\left(x\right)$ 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 equation $f\left(x\right)=0$ 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 $\frac{1}{2}$

$f\left(x\right)=x^2+ax+2;g\left(x\right)=x^2+2x+a$
Here a common root then
$\left|\begin{array}{cc}1& a\\ 1& 2\end{array}\right|\left|\begin{array}{cc}a& 2\\ 2& a\end{array}\right|={\left|\begin{array}{cc}2& 1\\ a& 1\end{array}\right|}^2$
$=a=2.-3$
$f\left(x\right)+g\left(x\right)=2x^2+(a+2)x+a+2$
Sum of roots $=\frac{-(a+2)}{2}$ if $a=-3$ then sum $=\frac{1}{2}$