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Question Number 154552 by 0731619 last updated on 19/Sep/21

Commented by Rasheed.Sindhi last updated on 20/Sep/21

•Is there any compitition to make posts of others ′red-flaged′ ? ♥Is there any compitition of ′likes′ of somebodies to their own posts? ⧫ Is this a market place? AND winners are those who have more ID′s or more divices!

$$\bullet\mathrm{Is}\:\mathrm{there}\:\mathrm{any}\:\mathrm{compitition}\:\mathrm{to}\:\mathrm{make} \\ $$$$\boldsymbol{\mathrm{posts}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{others}}\:'\mathrm{red}-\mathrm{flaged}'\:? \\ $$$$\heartsuit\mathrm{Is}\:\mathrm{there}\:\mathrm{any}\:\mathrm{compitition}\:\mathrm{of}\:'\mathrm{likes}' \\ $$$$\:\:\:\mathrm{of}\:\mathrm{somebodies}\:\mathrm{to}\:\boldsymbol{\mathrm{their}}\:\boldsymbol{\mathrm{own}}\:\boldsymbol{\mathrm{posts}}? \\ $$$$\blacklozenge\:\mathrm{Is}\:\mathrm{this}\:\mathrm{a}\:\boldsymbol{\mathrm{market}}\:\boldsymbol{\mathrm{place}}? \\ $$$$\boldsymbol{\mathrm{AND}}\: \\ $$$$\:\:\mathrm{winners}\:\mathrm{are}\:\mathrm{those}\:\mathrm{who}\:\mathrm{have}\:\mathrm{more} \\ $$$$\:\:\mathrm{ID}'\mathrm{s}\:\mathrm{or}\:\mathrm{more}\:\mathrm{divices}! \\ $$

Commented by Rasheed.Sindhi last updated on 19/Sep/21

@mathdanisur How did you guess that?

$$@{mathdanisur} \\ $$$$\mathcal{H}{ow}\:{did}\:{you}\:{guess}\:{that}? \\ $$

Commented by Tinku Tara last updated on 20/Sep/21

Answering own question is very much acceptable. Almost all senior mebers of forum post a quesion which they find interesting and answer in 2/3 days. Posting very simple question just increase noise and good challenging questions end up getting ignored which could create good discussion on solution approaches as well.

$$\mathrm{Answering}\:\mathrm{own}\:\mathrm{question}\:\mathrm{is}\:\mathrm{very}\:\mathrm{much} \\ $$$$\mathrm{acceptable}.\:\mathrm{Almost}\:\mathrm{all}\:\mathrm{senior}\:\mathrm{mebers}\: \\ $$$$\mathrm{of}\:\mathrm{forum}\:\mathrm{post}\:\mathrm{a}\:\mathrm{quesion}\:\mathrm{which}\:\mathrm{they} \\ $$$$\mathrm{find}\:\mathrm{interesting}\:\mathrm{and}\:\mathrm{answer}\:\mathrm{in}\:\mathrm{2}/\mathrm{3} \\ $$$$\mathrm{days}. \\ $$$$\mathrm{Posting}\:\mathrm{very}\:\mathrm{simple}\:\mathrm{question}\:\mathrm{just} \\ $$$$\mathrm{increase}\:\mathrm{noise}\:\mathrm{and}\:\mathrm{good}\:\mathrm{challenging} \\ $$$$\mathrm{questions}\:\mathrm{end}\:\mathrm{up}\:\mathrm{getting}\:\mathrm{ignored}\:\mathrm{which} \\ $$$$\mathrm{could}\:\mathrm{create}\:\mathrm{good}\:\mathrm{discussion}\:\mathrm{on}\:\mathrm{solution} \\ $$$$\mathrm{approaches}\:\mathrm{as}\:\mathrm{well}. \\ $$

Commented by liberty last updated on 19/Sep/21

Commented by liberty last updated on 19/Sep/21

What are you saying ?

Commented by iloveisrael last updated on 20/Sep/21

Sieht aus als wärst du verrückt

Commented by mathdanisur last updated on 20/Sep/21

and iloveisrael as well as that person

$$\mathrm{and}\:\:\boldsymbol{\mathrm{i}}\mathrm{loveisrae}\boldsymbol{\mathrm{l}}\:\:\mathrm{as}\:\mathrm{well}\:\mathrm{as}\:\mathrm{that}\:\mathrm{person} \\ $$

Commented by iloveisrael last updated on 20/Sep/21

In my opinion, let it be. do we feel harmed by that? what's the downside? We don't need to interfere in all our people's affairs. if it doesn't hurt us

Commented by Rasheed.Sindhi last updated on 20/Sep/21

Then it will be difficult for administator to distinguish between “really able to be red marked” and “marked red as a revenge”

$$\mathrm{Then}\:\mathrm{it}\:\mathrm{will}\:\mathrm{be}\:\mathrm{difficult}\: \\ $$$$\mathrm{for}\:\mathrm{administator}\:\mathrm{to}\:\mathrm{distinguish} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{between}\: \\ $$$$``{really}\:\boldsymbol{{able}}\:{to}\:{be}\:{red}\:{marked}'' \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{and}\: \\ $$$$``{marked}\:{red}\:{as}\:{a}\:\boldsymbol{{revenge}}'' \\ $$

Commented by mr W last updated on 20/Sep/21

forum is a public for exchanging questions & answers in mathematics. if somebody constantly posts pictures or other things which have nothing to do with exchanging mathematics, i don′t feel hurt, but disturbed.

$${forum}\:{is}\:{a}\:{public}\:{for}\:{exchanging} \\ $$$${questions}\:\&\:{answers}\:{in}\:{mathematics}. \\ $$$${if}\:{somebody}\:{constantly}\:{posts}\:{pictures} \\ $$$${or}\:{other}\:{things}\:{which}\:{have}\:{nothing} \\ $$$${to}\:{do}\:{with}\:{exchanging}\:{mathematics}, \\ $$$${i}\:{don}'{t}\:{feel}\:{hurt},\:{but}\:{disturbed}. \\ $$

Commented by mathdanisur last updated on 20/Sep/21

Ser, there are 3 or more profiles of this person and l mentioned it, he knowes who he and that person paints what l share in red, do his best, nothing will chang!

$$\boldsymbol{\mathrm{S}}\mathrm{er},\:\mathrm{there}\:\mathrm{are}\:\mathrm{3}\:\mathrm{or}\:\mathrm{more}\:\mathrm{profiles}\:\mathrm{of}\:\mathrm{this} \\ $$$$\mathrm{person}\:\mathrm{and}\:\mathrm{l}\:\mathrm{mentioned}\:\mathrm{it},\:\mathrm{he}\:\mathrm{knowes} \\ $$$$\mathrm{who}\:\mathrm{he}\:\mathrm{and}\:\mathrm{that}\:\mathrm{person}\:\mathrm{paints}\:\mathrm{what} \\ $$$$\mathrm{l}\:\mathrm{share}\:\mathrm{in}\:\mathrm{red},\:\mathrm{do}\:\mathrm{his}\:\mathrm{best},\:\mathrm{nothing}\:\mathrm{will} \\ $$$$\mathrm{chang}! \\ $$

Commented by liberty last updated on 20/Sep/21

try to explain if someone posted a question and then someone answered whether it was the same person, what were your losses and what bothered you.

Commented by mr W last updated on 20/Sep/21

if someone don′t want to exchange with others, he doesn′t need to use the forum function. people who use different identities to play the roles of questioner and answerer as well are cheating themselves. i don′t have any loss when other people cheat themselves, i just think it′s non−sense. or you don′t think so?

$${if}\:{someone}\:{don}'{t}\:{want}\:{to}\:{exchange}\: \\ $$$${with}\:{others},\:{he}\:{doesn}'{t}\:{need}\:{to}\:{use} \\ $$$${the}\:{forum}\:{function}.\:{people}\:{who}\:{use} \\ $$$${different}\:{identities}\:{to}\:{play}\:{the}\:{roles} \\ $$$${of}\:{questioner}\:{and}\:{answerer}\:{as}\:{well} \\ $$$${are}\:{cheating}\:{themselves}.\:\:{i}\:{don}'{t} \\ $$$${have}\:{any}\:{loss}\:{when}\:{other}\:{people}\: \\ $$$${cheat}\:{themselves},\:{i}\:{just}\:{think}\:{it}'{s} \\ $$$${non}−{sense}.\:{or}\:{you}\:{don}'{t}\:{think}\:{so}? \\ $$

Commented by mathdanisur last updated on 20/Sep/21

Ser, yes you are right, thanks.

$$\boldsymbol{\mathrm{S}}\mathrm{er},\:\mathrm{yes}\:\mathrm{you}\:\mathrm{are}\:\mathrm{right},\:\mathrm{thanks}. \\ $$

Commented by liberty last updated on 20/Sep/21

Mr. W, I noticed that you are the one who very often interferes in people's affairs, you position yourself as a traffic policeman who always determines whether people are right or wrong. those of you who always feel disturbed by posts like mister zainal. you are very great . 5 stars for you.

Commented by liberty last updated on 20/Sep/21

in this forum every member is free to post as long as what is posted does not offend or hurt us. let it be. After all, if you don't want to read it, don't read it. simple isn't it?

Commented by liberty last updated on 20/Sep/21

it's not a matter of reasonable or unreasonable. Don't make your measurements to measure others. the size we use for us personally. good for us is not necessarily good for others.

Commented by mr W last updated on 20/Sep/21

i don′t judge whether people are right or wrong. i raraly red flag other people′s posts. but when i see that some people post things which are not the original purpose of the forum, i just tell them my opinion. since the forum has no real administrator in duty, it depents on us all if the forum can remain clear. i personally have nothing against anybody in the forum.

$${i}\:{don}'{t}\:{judge}\:{whether}\:{people}\:{are}\:{right} \\ $$$${or}\:{wrong}.\:{i}\:{raraly}\:{red}\:{flag}\:{other} \\ $$$${people}'{s}\:{posts}.\:{but}\:{when}\:{i}\:{see}\:{that} \\ $$$${some}\:{people}\:{post}\:{things}\:{which}\:{are} \\ $$$${not}\:{the}\:{original}\:{purpose}\:{of}\:{the}\:{forum}, \\ $$$${i}\:{just}\:{tell}\:{them}\:{my}\:{opinion}.\:{since}\:{the}\: \\ $$$${forum}\:{has}\:{no}\:{real}\:{administrator} \\ $$$${in}\:{duty},\:{it}\:{depents}\:{on}\:{us}\:{all}\:{if}\:{the} \\ $$$${forum}\:{can}\:{remain}\:{clear}.\:{i}\:{personally} \\ $$$${have}\:{nothing}\:{against}\:{anybody}\:{in}\:{the} \\ $$$${forum}. \\ $$

Commented by mr W last updated on 20/Sep/21

if you let people do what has nothing to do with the forum, the forum will become a place full of carbage. do you want it be such a place?

$${if}\:{you}\:{let}\:{people}\:{do}\:{what}\:{has}\:{nothing} \\ $$$${to}\:{do}\:{with}\:{the}\:{forum},\:{the}\:{forum}\:{will} \\ $$$${become}\:{a}\:{place}\:{full}\:{of}\:{carbage}.\:{do} \\ $$$${you}\:{want}\:{it}\:{be}\:{such}\:{a}\:{place}? \\ $$

Commented by liberty last updated on 20/Sep/21

Dear Mr. W, again I say if there are members who post questions and no one answers then those who post provide answers, leave it alone, I'm sure someone will benefit from that answer. on the forum there are many statuses whether it's student, student or teacher or lecturer. Again I say don't make your size to measure others. problem member using 2 ID leave it alone. Don't get distracted with other people's business. if you are nosy it just shows you are arrogant in this forum. 5 stars for you

Commented by talminator2856791 last updated on 20/Sep/21

it is also annoying that someone answers their own questions.

$$\:\mathrm{it}\:\mathrm{is}\:\mathrm{also}\:\mathrm{annoying}\:\mathrm{that}\:\mathrm{someone}\: \\ $$$$\:\mathrm{answers}\:\mathrm{their}\:\mathrm{own}\:\mathrm{questions}. \\ $$

Commented by liberty last updated on 20/Sep/21

that′s his problem , want to post a question and he answers himself let it be the problem. if you feel annoying just keep it in your heart that resentful feeling. that′s your problem.

$${that}'{s}\:{his}\:{problem}\:,\:{want}\:{to} \\ $$$${post}\:{a}\:{question}\:{and}\:{he}\:{answers} \\ $$$${himself}\:{let}\:{it}\:{be}\:{the}\:{problem}. \\ $$$${if}\:{you}\:{feel}\:{annoying}\:{just}\:{keep} \\ $$$${it}\:{in}\:{your}\:{heart}\:{that}\:{resentful} \\ $$$${feeling}.\:{that}'{s}\:{your}\:{problem}. \\ $$

Commented by mr W last updated on 20/Sep/21

liberty sir: i decided not to discuss with you further about this matter. actually you didn′t understand and din′t really look at what i did. i never requested people to do things in the way i like. i never deleted other people′s posts because i don′t have the power to do that even i want! i even never thought that one may not answer his own questions. what i did is, when i see that somebody doesn′t use the forum properly i try to explain him what i think the forum is for. sometimes i try to give him concrete hints. all what i want is that the forum is a clean place where we like to be to exchange our love to mathematics. certainly you have the right to say that you don′t care about some posts which some other people don′t like. but you should also accept that these other people have the right not to like those posts. just image we have ten such persons like that guy who constantly posts his own home works in the forum and ignores all kinds of communication other people try with him. i think after a few days the forum will become a place full of carbage. i don′t want this happen, therefore i act. it′s your good right to say that i were arrogant. but i think the most people here may judge by themselve who is really arrogant.

$${liberty}\:{sir}:\:{i}\:{decided}\:{not}\:{to}\:{discuss} \\ $$$${with}\:{you}\:{further}\:{about}\:{this}\:{matter}. \\ $$$${actually}\:{you}\:{didn}'{t}\:{understand}\:{and} \\ $$$${din}'{t}\:{really}\:{look}\:{at}\:{what}\:{i}\:{did}.\:{i}\:{never} \\ $$$${requested}\:{people}\:{to}\:{do}\:{things}\:{in}\:{the} \\ $$$${way}\:{i}\:{like}.\:{i}\:{never}\:{deleted}\:{other} \\ $$$${people}'{s}\:{posts}\:{because}\:{i}\:{don}'{t}\:{have} \\ $$$${the}\:{power}\:{to}\:{do}\:{that}\:{even}\:{i}\:{want}! \\ $$$${i}\:{even}\:{never}\:{thought}\:{that}\:{one}\:{may}\: \\ $$$${not}\:{answer}\:{his}\:{own}\:{questions}. \\ $$$${what}\:{i}\:{did}\:{is},\:{when}\:{i}\:{see}\:{that}\:{somebody} \\ $$$${doesn}'{t}\:{use}\:{the}\:{forum}\:{properly}\:{i}\:{try} \\ $$$${to}\:{explain}\:{him}\:{what}\:{i}\:{think}\:{the} \\ $$$${forum}\:{is}\:{for}.\:{sometimes}\:{i}\:{try}\:{to}\:{give} \\ $$$${him}\:{concrete}\:{hints}.\:{all}\:{what}\:{i}\:{want} \\ $$$${is}\:{that}\:{the}\:{forum}\:{is}\:{a}\:{clean}\:{place}\: \\ $$$${where}\:{we}\:{like}\:{to}\:{be}\:{to}\:{exchange}\:{our} \\ $$$${love}\:{to}\:{mathematics}.\:{certainly}\:{you} \\ $$$${have}\:{the}\:{right}\:{to}\:{say}\:{that}\:{you}\:{don}'{t} \\ $$$${care}\:{about}\:{some}\:{posts}\:{which}\:{some} \\ $$$${other}\:{people}\:{don}'{t}\:{like}.\:{but}\:{you} \\ $$$${should}\:{also}\:{accept}\:{that}\:{these}\:{other} \\ $$$${people}\:{have}\:{the}\:{right}\:{not}\:{to}\:{like}\:{those} \\ $$$${posts}.\: \\ $$$${just}\:{image}\:{we}\:{have}\:{ten}\:{such}\:{persons} \\ $$$${like}\:{that}\:{guy}\:{who}\:{constantly}\:{posts}\: \\ $$$${his}\:{own}\:{home}\:{works}\:{in}\:{the}\:{forum}\: \\ $$$${and}\:{ignores}\:{all}\:{kinds}\:{of}\:{communication} \\ $$$${other}\:{people}\:{try}\:{with}\:{him}.\:{i}\:{think} \\ $$$${after}\:{a}\:{few}\:{days}\:{the}\:{forum}\:{will} \\ $$$${become}\:{a}\:{place}\:{full}\:{of}\:{carbage}.\:{i} \\ $$$${don}'{t}\:{want}\:{this}\:{happen},\:{therefore} \\ $$$${i}\:{act}. \\ $$$${it}'{s}\:{your}\:{good}\:{right}\:{to}\:{say}\:{that}\:{i}\:{were} \\ $$$${arrogant}.\:{but}\:{i}\:{think}\:{the}\:{most}\:{people}\: \\ $$$${here}\:{may}\:{judge}\:{by}\:{themselve}\:{who}\:{is} \\ $$$${really}\:{arrogant}. \\ $$

Commented by SLVR last updated on 26/Sep/21

Mr..W...sir...afew^ people cannot understand you +vely.. but i like that sacredness thanks...to your services... dear professor

$$ \\ $$$${Mr}..{W}...{sir}...{afe}\overset{} {{w}} \\ $$$${people}\:{cannot} \\ $$$${understand}\:{you}\:+{vely}.. \\ $$$${but}\:{i}\:{like}\:{that}\:{sacredness} \\ $$$${thanks}...{to}\:{your}\:{services}... \\ $$$${dear}\:{professor} \\ $$

Commented by mr W last updated on 26/Sep/21

thanks for understanding sir!

$${thanks}\:{for}\:{understanding}\:{sir}! \\ $$

Commented by Rasheed.Sindhi last updated on 26/Sep/21

Sir we know that you are not only an expert mathematician but also very sincere to the forum and its members.

$$\mathcal{S}{ir}\:{we}\:{know}\:{that}\:{you}\:{are}\:{not}\:{only} \\ $$$${an}\:\boldsymbol{{expert}}\:\boldsymbol{{mathematician}}\:{but} \\ $$$${also}\:\boldsymbol{{very}}\:\boldsymbol{{sincere}}\:{to}\:{the}\:\boldsymbol{{forum}} \\ $$$${and}\:\boldsymbol{{its}}\:\boldsymbol{{members}}. \\ $$

Commented by mr W last updated on 26/Sep/21

thanks sir! the same to you!

$${thanks}\:{sir}! \\ $$$${the}\:{same}\:{to}\:{you}! \\ $$

Commented by Rasheed.Sindhi last updated on 28/Sep/21

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Answered by MJS_new last updated on 19/Sep/21

squaring and transforming x^(10) +x^8 −2x^6 −2x^4 +1=0 (x^4 +x^2 −1)(x^6 −x^2 −1)=0 x^2 =−(1/2)±((√5)/2) ∨x^2 =(((1/2)−((√(69))/(18))))^(1/3) +(((1/2)+((√(69))/(18))))^(1/3) testing all solution I get for x∈R x=−(√(−(1/2)+((√5)/2)))∨x=(√((((1/2)−((√(69))/(18))))^(1/3) +(((1/2)+((√(69))/(18))))^(1/3) ))

$$\mathrm{squaring}\:\mathrm{and}\:\mathrm{transforming} \\ $$$${x}^{\mathrm{10}} +{x}^{\mathrm{8}} −\mathrm{2}{x}^{\mathrm{6}} −\mathrm{2}{x}^{\mathrm{4}} +\mathrm{1}=\mathrm{0} \\ $$$$\left({x}^{\mathrm{4}} +{x}^{\mathrm{2}} −\mathrm{1}\right)\left({x}^{\mathrm{6}} −{x}^{\mathrm{2}} −\mathrm{1}\right)=\mathrm{0} \\ $$$${x}^{\mathrm{2}} =−\frac{\mathrm{1}}{\mathrm{2}}\pm\frac{\sqrt{\mathrm{5}}}{\mathrm{2}}\:\vee{x}^{\mathrm{2}} =\sqrt[{\mathrm{3}}]{\frac{\mathrm{1}}{\mathrm{2}}−\frac{\sqrt{\mathrm{69}}}{\mathrm{18}}}+\sqrt[{\mathrm{3}}]{\frac{\mathrm{1}}{\mathrm{2}}+\frac{\sqrt{\mathrm{69}}}{\mathrm{18}}} \\ $$$$\mathrm{testing}\:\mathrm{all}\:\mathrm{solution}\:\mathrm{I}\:\mathrm{get}\:\mathrm{for}\:{x}\in\mathbb{R} \\ $$$${x}=−\sqrt{−\frac{\mathrm{1}}{\mathrm{2}}+\frac{\sqrt{\mathrm{5}}}{\mathrm{2}}}\vee{x}=\sqrt{\sqrt[{\mathrm{3}}]{\frac{\mathrm{1}}{\mathrm{2}}−\frac{\sqrt{\mathrm{69}}}{\mathrm{18}}}+\sqrt[{\mathrm{3}}]{\frac{\mathrm{1}}{\mathrm{2}}+\frac{\sqrt{\mathrm{69}}}{\mathrm{18}}}} \\ $$

Answered by liberty last updated on 19/Sep/21

x^3 −(1/x) = (1/( (√(x^2 +1)))) let x=tan θ ⇒tan^3 θ−cot θ=(1/(sec θ)) ⇒tan^3 θ−cot θ=cos θ ⇒((sin^3 θ)/(cos^3 θ))−((cos θ)/(sin θ))=cos θ ⇒sin^4 θ−cos^4 θ=cos^4 θ sin θ ⇒2cos^2 θ−1=cos^4 θ sin θ ⇒2cos^2 θ−1=cos^4 θ (√(1−cos^2 θ)) ⇒2q−1=q^2 (√(1−q )) , q=cos^2 θ ⇒4q^2 −4q+1=q^4 −q^5 ⇒q^5 −q^4 +4q^2 −4q+1=0 aproximation { ((q≈0.430159=cos^2 θ)),((q≈0.618033=cos^2 θ)) :}

$$\:{x}^{\mathrm{3}} −\frac{\mathrm{1}}{{x}}\:=\:\frac{\mathrm{1}}{\:\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}} \\ $$$$\:{let}\:{x}=\mathrm{tan}\:\theta\: \\ $$$$\Rightarrow\mathrm{tan}\:^{\mathrm{3}} \theta−\mathrm{cot}\:\theta=\frac{\mathrm{1}}{\mathrm{sec}\:\theta} \\ $$$$\Rightarrow\mathrm{tan}\:^{\mathrm{3}} \theta−\mathrm{cot}\:\theta=\mathrm{cos}\:\theta \\ $$$$\Rightarrow\frac{\mathrm{sin}\:^{\mathrm{3}} \theta}{\mathrm{cos}\:^{\mathrm{3}} \theta}−\frac{\mathrm{cos}\:\theta}{\mathrm{sin}\:\theta}=\mathrm{cos}\:\theta \\ $$$$\Rightarrow\mathrm{sin}\:^{\mathrm{4}} \theta−\mathrm{cos}\:^{\mathrm{4}} \theta=\mathrm{cos}\:^{\mathrm{4}} \theta\:\mathrm{sin}\:\theta \\ $$$$\Rightarrow\mathrm{2cos}\:^{\mathrm{2}} \theta−\mathrm{1}=\mathrm{cos}^{\mathrm{4}} \theta\:\mathrm{sin}\:\theta \\ $$$$\Rightarrow\mathrm{2cos}\:^{\mathrm{2}} \theta−\mathrm{1}=\mathrm{cos}\:^{\mathrm{4}} \theta\:\sqrt{\mathrm{1}−\mathrm{cos}\:^{\mathrm{2}} \theta}\: \\ $$$$\Rightarrow\mathrm{2}{q}−\mathrm{1}={q}^{\mathrm{2}} \:\sqrt{\mathrm{1}−{q}\:}\:,\:{q}=\mathrm{cos}\:^{\mathrm{2}} \theta \\ $$$$\Rightarrow\mathrm{4}{q}^{\mathrm{2}} −\mathrm{4}{q}+\mathrm{1}={q}^{\mathrm{4}} −{q}^{\mathrm{5}} \\ $$$$\Rightarrow{q}^{\mathrm{5}} −{q}^{\mathrm{4}} +\mathrm{4}{q}^{\mathrm{2}} −\mathrm{4}{q}+\mathrm{1}=\mathrm{0} \\ $$$${aproximation} \\ $$$$\:\begin{cases}{{q}\approx\mathrm{0}.\mathrm{430159}=\mathrm{cos}\:^{\mathrm{2}} \theta}\\{{q}\approx\mathrm{0}.\mathrm{618033}=\mathrm{cos}\:^{\mathrm{2}} \theta}\end{cases} \\ $$$$ \\ $$

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