Question

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

Answer 1

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

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