Question

'$af\left(k\right)<0$' is the necessary and sufficient condition for a particular real number $k$ to lie between the roots of a quadratic equation $f\left(x\right)=0$, where $f\left(x\right)=a{x}^2+bx+c$. If $f\left(k_1\right)f\left(k_2\right)<0$, then exactly one of the roots will lie between $k_1$ and $k_2$.

If $a(a+b+c)<0<c(a+b+c)$, then

• A. one root is less than $0$, the other is greater than $1$
• B. exactly one of the roots lies in $(0,1)$
• C. both the roots lie in $(0,1)$
• D. at least one of the roots lies in $(0,1)$

Answer 1

The correct option is A one root is less than $0$, the other is greater than $1$

$a(a+b+c)<0<c(a+b+c)$
$af\left(1\right)<0$
So, $1$ lies between the roots.
$f\left(0\right)f\left(1\right)>0$
So, no roots lies between $0$ and $1$.

Therefore, one root is greater than $1$ and another is less than $0$.