Question

Two distinct polynomial f(x) and g(x) are defined as follows:

$f\left(x\right)=x^2+ax+2;g\left(x\right)=x^2+2x+a$

If the equation f (x) = 0 and g(x) = 0 have a common root, then the sum of the roots of the equation f (x) + g(x) = 0 is

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

Answer 1

The correct option is D

$\frac{1}{2}$

f (x) = x${}^2$ + ax + 2 g(x) = 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(x) + g(x) = 2x${}^2$ + (a + 2)x + a + 2

Sum of roots = $\frac{-1(a+2)}{2}$ if a = - 3 then sum = $\frac{1}{2}$