Question

The number of integral values of $a$ such that the difference between the roots of the equation $x^2+ax-a=0$ is less than $1,$ is

• A. $2$
• B. $3$
• C. $4$
• D. $5$

Answer 1

The correct option is D $5$

$x^2+ax-a=0$
Let $\alpha ,\beta (\alpha >\beta )$ be the roots.
Then, $\alpha +\beta =-a$ and $\alpha \beta =-a$
Now, $\alpha -\beta <1$
$\Rightarrow \sqrt{(\alpha +\beta {)}^2-4\alpha \beta }<1$
$\Rightarrow a^2+4a-1<0$
$\Rightarrow (a+2{)}^2-5<0$
$\Rightarrow (a+2+\sqrt{5})(a+2-\sqrt{5})<0$
$\Rightarrow a\in (-2-\sqrt{5},\sqrt{5}-2)$
Possible integral values of $a$ are $-4,-3,-2,-1,0$