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

Question

Dr. Abubacker · Accepted Answer

(a)
Let $\alpha$ and $\beta$ be the roots of given equation $x^{2} + ax + 1 = 0$
Then
$\alpha + \beta = - a$ and $\alpha\beta = 1$
Now, $|\alpha - \beta| = \sqrt{(\alpha + \beta)^{2} - 4\alpha\beta} = \sqrt{a^{2} - 4}$
Given condition, $\sqrt{a^{2} - 4} < \ \sqrt{5}$
$\Rightarrow a^{2} - 4 < 5\ \  \Rightarrow \ \ \ |a| < 3$
$\Rightarrow a \in ( - 3,\ 3)$

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

Explanation

Reviewed by Dr. Abubacker

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