find all irational numbers x such that x 2+20x+20 and x^(3 ) −2020x+1 both is a rasional number.


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Question Number 114965 by joki last updated on 22/Sep/20

$f i n d a l l i r a t i o n a l n u m b e r s x s u c h t h a t x$ $2 + 20 x + 20 a n d x^{3} - 2020 x + 1 b o t h i s a$ $r a s i o n a l n u m b e r .$
	Answered by MJS_new last updated on 22/Sep/20

	$(1) x^{2} + 20 x + 20 \in Q$ $(2) x^{3} - 2020 x + 1 \in Q$ $x = a \pm \sqrt{b}; a, b \in Q; b \notin Q$ $(1)$ $\pm 2 (a + 10) \sqrt{b} + a^{2} + 20 a + b + 20 \in Q$ $\Leftrightarrow$ $2 (a + 10) \sqrt{b} \in Q$ $(2)$ $\pm (3 a^{2} + b - 2020) \sqrt{b} + a^{3} + 3 a b - 2020 a + 1 \in Q$ $\Leftrightarrow$ $(3 a^{2} + b - 2020) \sqrt{b} \in Q$ $now we have$ $(1) 2 (a + 10) \sqrt{b} \in Q$ $(2) (3 a^{2} + b - 2020) \sqrt{b} \in Q$ $\Rightarrow$ $either b = 0 but then x is not irrational$ $or a + 10 = 0 \land 3 a^{2} + b - 2020 = 0$ $\Rightarrow a = - 10 \land b = 1720$ $\Rightarrow x = - 10 \pm 2 \sqrt{430}$
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