Question

The integral value of $a$ , for which the quadratic expression $a{x}^2+\left(a-2\right)x-2$ is negative for exactly two integral values of $x,$ is

Answer 1

Answer: 1

Let, $f\left(x\right)=a{x}^2+\left(a-2\right)x-2$
$\therefore f\left(0\right)=-2$ and $f\left(-1\right)=0$
Since, the quadratic expression is negative for exactly two integral values
$\Rightarrow f\left(1\right)<0$ and $f\left(2\right)\ge 0$

$\Rightarrow a+a-2-2<0$ and $4a+2a-4-2\ge 0$
$\Rightarrow a<2$ and $a\ge 1$
$\Rightarrow a\in [1,2)$
Hence, the integral value of $a=1$