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For which of the sets of a, is the statement, “The difference between the roots of the equation $x^2+ax+1 = $ is less than $ \sqrt 5”$ true?

$\begin {array} {1 1} (A)\;(-3,\infty) & \quad (B)\;(3,\infty) \\ (C)\;(-\infty, -3) & \quad (D)\;(-3, 3) \end {array}$

Can you answer this question?
 
 

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Ans : (D)
Let $ \alpha,\: \beta$ be the roots of the eqn. $x^2+ax+1=0$, then
$ \alpha+\beta = -a\: and \: \alpha\beta = 1$
Now $| \alpha – \beta | = \sqrt{[( \alpha + \beta)2 - 4 \alpha\beta]}$
$ \Rightarrow | \alpha – \beta | = \sqrt{a^2} – 4$
According to the question,
$ \sqrt{a^2} – 4 < \sqrt5$
$ \Rightarrow a ^2 – 4 < 5$
$ \Rightarrow a^2 – 9 < 0$
$ \Rightarrow a \in (-3, 3)$
answered Jan 24, 2014 by thanvigandhi_1
 

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