(a , 2a +1 ]∩[ a^( 2) −a , a^( 2) + 4a +1 )≠ ∅ a ∈ ?


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Question Number 147487 by mnjuly1970 last updated on 21/Jul/21

$(a, 2 a + 1] \cap [a^{2} - a, a^{2} + 4 a + 1) \neq \emptyset$ $a \in ?$
	Answered by Rasheed.Sindhi last updated on 21/Jul/21

	$∙ a < 2 a + 1 \land a^{2} - a < a^{2} + 4 a + 1$ $a > - 1 \land a > - 1 / 5 \Rightarrow a > - 1 / 5 = - 0.2$ $∙ 2 a + 1 ⩾ a^{2} - a [∵ I n t e r s e c t i o n i s e m p t y]$ $a^{2} - 3 a - 1 ⩾ 0$ $a ⩾ \frac{3 \pm \sqrt{9 + 4}}{2} = \frac{3 \pm \sqrt{13}}{2}$ $∵ \frac{3 - \sqrt{13}}{2} ≯ - 1 / 5 ∴ R e j e c t e d$ $∴ a ⩾ \frac{3 + \sqrt{13}}{2}$ $a \in [\frac{3 + \sqrt{13}}{2}, \infty)$
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