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

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

Answered by MJS_new last updated on 22/Sep/20

(1) x^2 +20x+20∈Q  (2) x^3 −2020x+1∈Q    x=a±(√b); a, b ∈Q; b∉Q    (1)  ±2(a+10)(√b)+a^2 +20a+b+20∈Q  ⇔  2(a+10)(√b)∈Q    (2)  ±(3a^2 +b−2020)(√b)+a^3 +3ab−2020a+1∈Q  ⇔  (3a^2 +b−2020)(√b)∈Q    now we have  (1) 2(a+10)(√b)∈Q  (2) (3a^2 +b−2020)(√b)∈Q  ⇒  either b=0 but then x is not irrational  or a+10=0∧3a^2 +b−2020=0  ⇒ a=−10∧b=1720  ⇒ x=−10±2(√(430))