Question

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

• A. $(−3, 3)$
• B. $(−3, ∞)$
• C. $(3, ∞)$
• D. $(−∞, −3)$

Answer 1

Answer: (A)

$x^{2} + ax + 1 = 0$
$α + β = −a\, αβ =1$
$\left|α-β\right|=\sqrt{\left(\alpha+\beta\right)^{2}-4\alpha\beta}$
$\left|α-β\right|=\sqrt{a^{2}-4}$
$\sqrt{a^{2}-4} < \sqrt{5}$
$a^{2}-4 < 5
a^{2}-9 < 0$
$a\,\in\left(-3, 3\right).$