First slide

Theory of expressions

Question

If the absolute value of the difference of the roots of the equation $x^{2} + a x + 1 = 0$ exceeds $\sqrt{3 a},$ then

Moderate

A

$a \in (- \infty, - 1) \cup (4, \infty)$
B

$a \in (4, \infty)$
C

$a \in (- 1, 4)$
D

$a \in [0, 4)$

Solution

We have,
$\begin{array}{l} | α - β | > \sqrt{3 a} \\ \Rightarrow | α - β |^{2} > 3 a \\ \Rightarrow (α + β)^{2} - 4 α β > 3 a \\ \Rightarrow a^{2} - 4 > 3 a \\ \Rightarrow a^{2} - 3 a - 4 > 0 \Rightarrow (a - 4) (a + 1) > 0 \\ \Rightarrow a \in (- \infty, - 1) \cup (4, \infty) \end{array}$

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