Question Number 148689 by liberty last updated on 30/Jul/21

Solve for x∈R ((x^2 −x+1)/(x^2 +x+1)) = (√((x^3 −1)/(x^3 +1))) .

Answered by MJS_new last updated on 30/Jul/21

squaring (beware of false solutions!) and transforming leads to x^6 +(1/2)x^4 −(3/2)x^2 −(1/2)=0 let x=±(√(z−(1/6))) z^3 −((19)/(12))z−((13)/(54))=0 z_1 =−((√(19))/3)cos ((π/6)+(1/3)arcsin ((26(√(19)))/(361))) ≈−1.17399865514 z_2 =−((√(19))/3)sin ((1/3)arcsin ((26(√(19)))/(361))) ≈−.154370149574 z_3 =((√(19))/3)sin ((π/3)+(1/3)arcsin ((26(√(19)))/(361))) ≈1.32836880471 ⇒ x_1 ≈±1.15787102987i x_2 ≈±.566601108577i x_3 ≈±1.07782286951 testing shows only x_2 and x_3 solve the given equation. since x∈R we get x≈±1.07782286951

Commented byliberty last updated on 30/Jul/21

Commented byliberty last updated on 30/Jul/21

correct sir. thanks

Answered by Rasheed.Sindhi last updated on 30/Jul/21

((x^2 −x+1)/(x^2 +x+1)) = (√((x^3 −1)/(x^3 +1))) ((√((x^2 −x+1)/(x^2 +x+1))))^2 = (√(((x−1)(x^2 +x+1))/((x+1)(x^2 −x+1)))) ((√((x^2 −x+1)/(x^2 +x+1))))^2 = (√((x−1)/(x+1))).(√((x^2 +x+1)/(x^2 −x+1))) ((√((x^2 −x+1)/(x^2 +x+1))))^2 − (√((x−1)/(x+1))).((√((x^2 −x+1)/(x^2 +x+1))))^(−1) =0 (√((x^2 −x+1)/(x^2 +x+1)))((√((x^2 −x+1)/(x^2 +x+1)))−(√((x−1)/(x+1))).((√((x^2 −x+1)/(x^2 +x+1))))^(−2) )=0 (√((x^2 −x+1)/(x^2 +x+1)))=0 ∣ (√((x^2 −x+1)/(x^2 +x+1)))=(√((x−1)/(x+1)))((√((x^2 +x+1)/(x^2 −x+1))))^2 ((x^2 −x+1)/(x^2 +x+1))=((x−1)/(x+1))(((x^2 +x+1)/(x^2 −x+1)))^2 (x+1)(x^2 −x+1)^3 =(x−1)(x^2 +x+1)^3 (x^3 +1)(x^2 −x+1)^2 =(x^3 −1)(x^2 +x+1)^2 ..... ....

Commented byRasheed.Sindhi last updated on 30/Jul/21

Sir MJS, please guide me where I lose real roots?

Commented byMJS_new last updated on 30/Jul/21

your mistake is between lines 4 and 5 line 4: ((√a))^2 −(√b)(√a^(−1) )=0 ⇒^(???) line 5: (√a^(−1) )((√a^(−1) )−(√b))=0

Commented byRasheed.Sindhi last updated on 30/Jul/21

Yes sir, ThanX a Lot!

Commented byMJS_new last updated on 30/Jul/21

you′re welcome as always!

Commented byRasheed.Sindhi last updated on 31/Jul/21

Sir, one thing I feel that nowadays you aren′t contributing very much?

Commented byajfour last updated on 31/Jul/21

i had not an idea, it had to do anything with the Riemann thing, that question just surfaced on my mind. thanks for the link. i have bookmarked it, indefinite postponement.

Commented byMJS_new last updated on 31/Jul/21

to Sir Rasheed Sindhi: I feel the quality of questions has been dropping. I stopped answering certain people because I don′t do their homework for free and on the other hand there are those who ask for the solution of Riemann′s Conjecture on one day and for the result of simple fractional arithmetics the other day...

Commented byajfour last updated on 31/Jul/21

the ssymptote thing we are sure i can do myself, again sorry to have asked you that.

Commented byMJS_new last updated on 31/Jul/21

I didn′t mean to upset you, as you are one of the few persons who are the reason I′m still here. asymptotes of what? I guess I missed something, sorry

Commented byRasheed.Sindhi last updated on 01/Aug/21

to Sir MJ^(⧫) S: You′re right sir! I think there are: 4 catagories of forum-members (A)Some wants some help: They do math themselves also. They want help when they don′t know or they want variety of solutions or they want to upgrade their understanding of math.... They also give feedback(positive/ negative), thank/appreciate to the persons who have provided some kind of asistance to them. (B)Some others want others to do whole work fo their assinments. They don′t try themselves.Even they don′t want to learn much.They only use others′ work.... (C)Still others only wants to engage others.They don′t have learning goal. (Who one day ask high level questions and other day ask too simple questions as you mentioned) ... (D)Who want to help others. Help A-catagory .For them you must be always here.They want to learn sincerely from you sir!