First slide

Theory of equations

Question

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

Moderate

A

$(3, 5)$
B

$(- \infty, - 3)$
C

$(- 3, 3)$
D

$(- 3, \infty)$

Solution

Let $α$ and $β$ be the roots of the equation $x^{2} + a x + 1 = 0$
then, $α + β = - a$ and $α β = 1$
It is given that
$\begin{array}{l} | α - β | < \sqrt{5} \\ \Rightarrow (α - β)^{2} < 5 \\ \Rightarrow (α + β)^{2} - 4 α β < 5 \\ \Rightarrow a^{2} - 4 < 5 \Rightarrow a^{2} - 9 < 0 \Rightarrow - 3 < a < 3 \end{array}$

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