Question Number 148229 by iloveisrael last updated on 26/Jul/21

Commented bynimnim last updated on 26/Jul/21

x+(√(x^2 −1))+(1/(x−(√(x^2 −1))))=20 ⇒((x^2 −(x^2 −1)+1)/(x−(√(x^2 −1))))×((x+(√(x^2 −1)))/(x+(√(x^2 −1))))=20 ⇒((2(x+(√(x^2 −1))))/(x^2 −(x^2 −1)))=20 ⇔ x+(√(x^2 −1))=10 ⇒x^2 −1=100−20x+x^2 ⇔ x=((101)/(20)) Now, x^2 +(√(x^4 −1))+(1/(x^2 +(√(x^4 −1)))) = x^2 +(√(x^4 −1))+(1/(x^2 +(√(x^4 −1))))×((x^2 −(√(x^4 −1)))/(x^2 −(√(x^4 −1)))) =x^2 +(√(x^4 −1))+((x^2 −(√(x^4 −1)))/(x^4 −(x^4 −1))) =x^2 +(√(x^4 −1))+x^2 −(√(x^4 −1)) =2x^2 =2(((101)/(20)))^2 =((10201)/(200))=51.005★

Answered by EDWIN88 last updated on 26/Jul/21

(1)(x+(√(x^2 −1)))(x−(√(x^2 −1)))+1=20(x−(√(x^2 −1))) ⇒2 = 20(x−(√(x^2 −1))) ⇒(√(x^2 −1)) = x−(1/(10)) ⇒x^2 −1= x^2 −(x/5)+(1/(100)) ⇒(x/5)=((101)/(100)) , x=((101)/(20)) (2)x^2 +(√(x^4 −1)) +(1/(x^2 +(√(x^4 −1)))) =? (•) x^2 −1+(√((x^2 −1)(x^2 +1)))+1 = (x+1)(x−1)+(√((x+1)(x−1)(x^2 +1)))+1 = (((121)/(20)))(((81)/(20)))+(√((((121)/(20)))(((81)/(20)))(((101^2 +20^2 )/(20^2 )))))+1 = ((9801)/(400))+((99(√(10601)))/(400)) +1 =((99(√(10601))+10201)/(400)) so x^2 +(√(x^4 −1))+(1/(x^2 +(√(x^4 −1)))) = ((99(√(10601))+10201)/(400)) +((400)/( 99(√(10601))+10201)) [ love Jew ]