Question 3: An investor’s return \(R(x)\) in rupees after investing \(x\) thousand rupees is given by quadratic expression:
\[R(x)=-x^2+6x-8\]
Factorize the expression and find the investment levels that result in zero return.
Solution:
$R(x)=-x^{2} +6x-8$
$\ \ \ \ \ \ \ =-x^{2} +4x+2x-8$
$\ \ \ \ \ \ \ =x( -x +4) -2( -x+4)$
$\ \ \ \ \ \ \ =( -x +4)( x-2)$
For zero investement level $\displaystyle R( x) =0$
$( -x +4)( x-2) =0$
$-x +4=0\ \ \ \ \ \ \ \ \ |\ \ \ \ \ \ \ \ \ \ \ \ \ x-2=0$
$x =4\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ |\ \ \ \ \ \ \ \ \ \ \ \ \ x=2$
Investement level that result in zero return will be at $x=2$ and $x=4$.