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Question Number 190972 by pascal889 last updated on 15/Apr/23

Answered by Frix last updated on 15/Apr/23

(1/(2−(√3)))=2+(√3)∧(1/(2+(√3)))=2−(√3) a=2∧b=(√3) (a+b)^2 +(a−b)^2 =2(a^2 +b^2 )=14

$$\frac{\mathrm{1}}{\mathrm{2}−\sqrt{\mathrm{3}}}=\mathrm{2}+\sqrt{\mathrm{3}}\wedge\frac{\mathrm{1}}{\mathrm{2}+\sqrt{\mathrm{3}}}=\mathrm{2}−\sqrt{\mathrm{3}} \\ $$$${a}=\mathrm{2}\wedge{b}=\sqrt{\mathrm{3}} \\ $$$$\left({a}+{b}\right)^{\mathrm{2}} +\left({a}−{b}\right)^{\mathrm{2}} =\mathrm{2}\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} \right)=\mathrm{14} \\ $$

Answered by Rasheed.Sindhi last updated on 15/Apr/23

AnOther way m=(1/(2−(√3) ))=2+(√3) n=(1/(2+(√3)))=2−(√3) m^2 +n^2 =(m+n)^2 −2mn =(2+(√3) +2−(√3) )^2 −2(2+(√3) )(2−(√3) ) (4)^2 −2(4−3)=16−2=14

$$\mathrm{AnOther}\:\mathrm{way} \\ $$$${m}=\frac{\mathrm{1}}{\mathrm{2}−\sqrt{\mathrm{3}}\:}=\mathrm{2}+\sqrt{\mathrm{3}}\: \\ $$$${n}=\frac{\mathrm{1}}{\mathrm{2}+\sqrt{\mathrm{3}}}=\mathrm{2}−\sqrt{\mathrm{3}}\: \\ $$$${m}^{\mathrm{2}} +{n}^{\mathrm{2}} =\left({m}+{n}\right)^{\mathrm{2}} −\mathrm{2}{mn} \\ $$$$\:\:\:\:\:=\left(\mathrm{2}+\cancel{\sqrt{\mathrm{3}}\:}+\mathrm{2}−\cancel{\sqrt{\mathrm{3}}\:}\:\right)^{\mathrm{2}} −\mathrm{2}\left(\mathrm{2}+\sqrt{\mathrm{3}}\:\right)\left(\mathrm{2}−\sqrt{\mathrm{3}}\:\right) \\ $$$$\left(\mathrm{4}\right)^{\mathrm{2}} −\mathrm{2}\left(\mathrm{4}−\mathrm{3}\right)=\mathrm{16}−\mathrm{2}=\mathrm{14} \\ $$

Commented by Acem last updated on 16/Apr/23

Nice mon ami Rasheed So long i didn′t come Hope you and all are okey

$${Nice}\:\:{mon}\:{ami}\:{Rasheed} \\ $$$$ \\ $$$$\:{So}\:{long}\:{i}\:{didn}'{t}\:{come} \\ $$$$ \\ $$$$\:{Hope}\:{you}\:{and}\:{all}\:{are}\:{okey} \\ $$$$ \\ $$

Commented by Rasheed.Sindhi last updated on 16/Apr/23

Welcome sir Acem! Missed you a lot, during your long term absence... I remember your exciting comments... BTW, where were you in this period sir?

$${Welcome}\:{sir}\:{Acem}! \\ $$$${Missed}\:{you}\:{a}\:{lot},\:{during}\:\:{your} \\ $$$$\:{long}\:{term}\:{absence}... \\ $$$${I}\:{remember}\:{your}\:{exciting}\:{comments}... \\ $$$$\: \\ $$$${BTW}, \\ $$$$\:{where}\:{were}\:{you}\:{in}\:{this}\:{period}\:{sir}? \\ $$

Commented by Acem last updated on 17/Apr/23

I missed you too, am glad that you and all are okey with all my heart! My country is still in a war, i lost all my books, plans and my own studies five years ago. Untill now i haven′t been able to settle in a safe place. One day, i will restore all my knowledge. Life is cruel and my soul knows no despair.

$$\:{I}\:{missed}\:{you}\:{too},\:{am}\:{glad}\:{that}\:{you}\:{and}\:{all} \\ $$$${are}\:{okey}\:{with}\:{all}\:{my}\:{heart}! \\ $$$$ \\ $$$${My}\:{country}\:{is}\:{still}\:{in}\:{a}\:{war},\:{i}\:{lost}\:{all}\:{my}\: \\ $$$${books},\:{plans}\:{and}\:{my}\:{own}\:{studies}\:{five}\:{years} \\ $$$${ago}.\:{Untill}\:{now}\:{i}\:{haven}'{t}\:{been}\:{able}\:{to}\:{settle} \\ $$$${in}\:{a}\:{safe}\:{place}. \\ $$$$ \\ $$$${One}\:{day},\:{i}\:{will}\:{restore}\:{all}\:{my}\:{knowledge}. \\ $$$${Life}\:{is}\:{cruel}\:{and}\:{my}\:{soul}\:{knows}\: \\ $$$${no}\:{despair}. \\ $$$$ \\ $$

Commented by manxsol last updated on 17/Apr/23

Bienvenido, Sir Acem, 104 dias sin su sabiduria y sus buenos consejos.Que todo mejore para Usted .

$${Bienvenido},\:{Sir}\:{Acem},\:\mathrm{104}\: \\ $$$${dias}\:{sin}\:{su}\:{sabiduria}\:{y}\:{sus} \\ $$$$\:{buenos}\:{consejos}.{Que}\:{todo} \\ $$$$\:{mejore}\:{para}\:{Usted}\:. \\ $$

Commented by Rasheed.Sindhi last updated on 17/Apr/23

@Acem Very sad to hear about your circumstances...btw which country do you belong to?... Well wishes about your future sir!

$$@{Acem} \\ $$$${Very}\:{sad}\:{to}\:{hear}\:{about}\:{your}\: \\ $$$${circumstances}...{btw}\:{which}\:{country} \\ $$$${do}\:{you}\:{belong}\:{to}?...\:{Well}\:{wishes} \\ $$$${about}\:{your}\:{future}\:{sir}!\: \\ $$

Commented by Acem last updated on 22/Apr/23

@Mr. Manxsol Bienvenido :) Como esta? Espero que estes bien. Pido disculpas por esta ausencia involuntaria. Esperamos dias mejores. La ciencia que sirve a la humanidad es debil e impotente ante la ciencia privada que sirve a los demonios inmorales del mundo Somos las unicas victimas, y nada mas para blanquear el rostro de la civilizacion

$$\left.@{Mr}.\:{Manxsol}\:\:{Bienvenido}\::\right)\:{Como}\:{esta}? \\ $$$${Espero}\:{que}\:{estes}\:{bien}. \\ $$$${Pido}\:{disculpas}\:{por}\:{esta}\:{ausencia}\:{involuntaria}. \\ $$$${Esperamos}\:{dias}\:{mejores}. \\ $$$$ \\ $$$${La}\:{ciencia}\:{que}\:{sirve}\:{a}\:{la}\:{humanidad}\:{es}\:{debil}\:{e}\: \\ $$$${impotente}\:{ante}\:{la}\:{ciencia}\:{privada} \\ $$$${que}\:{sirve}\:{a}\:{los}\:{demonios}\:{inmorales}\:{del}\:{mundo} \\ $$$$ \\ $$$${Somos}\:{las}\:{unicas}\:{victimas},\:{y}\:{nada}\:{mas}\:{para} \\ $$$${blanquear}\:{el}\:{rostro}\:{de}\:{la}\:{civilizacion} \\ $$

Commented by Acem last updated on 22/Apr/23

@Sir Rasheed, thank you so mush :) for your wishes i need. Syria

$$\left.@{Sir}\:{Rasheed},\:{thank}\:{you}\:{so}\:{mush}\::\right) \\ $$$${for}\:{your}\:{wishes}\:{i}\:{need}. \\ $$$$ \\ $$$${Syria} \\ $$$$ \\ $$

Commented by Rasheed.Sindhi last updated on 22/Apr/23

Thanks sir Acem! May God protect you and your country from war!

$$\mathcal{T}{hanks}\:{sir}\:{Acem}! \\ $$$${May}\:{God}\:{protect}\:{you}\:{and}\:{your}\: \\ $$$${country}\:{from}\:{war}! \\ $$

Commented by Acem last updated on 23/Apr/23

Big thanks sir Rasheed! May Allah bless you, Amin for all

$${Big}\:{thanks}\:{sir}\:{Rasheed}! \\ $$$${May}\:{Allah}\:{bless}\:{you},\: \\ $$$$\:{Amin}\:{for}\:{all} \\ $$

Commented by Rasheed.Sindhi last updated on 25/Apr/23

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Answered by manxsol last updated on 16/Apr/23

m+n=4 mn=1 m^2 +n^2 =(4)^2 −2=14

$${m}+{n}=\mathrm{4}\:{mn}=\mathrm{1} \\ $$$${m}^{\mathrm{2}} +{n}^{\mathrm{2}} =\left(\mathrm{4}\right)^{\mathrm{2}} −\mathrm{2}=\mathrm{14} \\ $$

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