Question

If the difference between the roots of the equation $x^2+ax+1=0$ is less than $\sqrt{5}$, then choose the set of possible values of $a$ is:

• A. $\{-2,1\}$
• B. $\{-4,3\}$
• C. $\{-4,4\}$
• D. $\{-6,1\}$

Answer 1

The correct option is A $\{-2,1\}$

Let $p,q$ be the roots of the equation $x^2+ax+1=0$ and
$\Rightarrow |p-q|<\sqrt{5}$
Now, using the conidition for sum and product of roots, we get:
$p+q=-a\&p\times q=1$

$\Rightarrow \sqrt{(p+q{)}^2-4pq}<\sqrt{5}$

$\Rightarrow \sqrt{a^2-4}<\sqrt{5}$
$\Rightarrow a^2-4<5$
$\Rightarrow a^2<9$
$\Rightarrow -3<a<3$
$\therefore a\in (-3,3)$