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Question Number 7077 by Tawakalitu. last updated on 09/Aug/16

Commented by Yozzii last updated on 09/Aug/16

a^2 +a=5 Let u=a+3⇒a=u−3 ⇒(u−3)^2 +u−3=5 u^2 −6u+9+u−3=5 u^2 −5u+6=5 u^2 −5u+1=0 u=((5±(√(25−4×1)))/2)=((5±(√(21)))/2) If u=((5+(√(21)))/2) u^(−1) =(2/(5+(√(21))))=((2(5−(√(21))))/(25−21))=((5−(√(21)))/2) ∴ u^3 +u^(−3) =(1/2^3 )((5+(√(21)))^3 +(5−(√(21)))^3 ) =(1/8)(5+(√(21))+5−(√(21)))(5^2 +21+10(√(21))−10(√(21))+5^2 +21+−25+21) =(5/4)(50+42−4) =(5/4)(88) =5×22 =110 (a+3)^3 +(a+3)^(−3) =110 If u=((5−(√(21)))/2)⇒u^(−1) =((5+(√(21)))/2) ⇒u^3 +u^(−3) =110 or (a+3)^3 +(a+3)^(−3) =110

$${a}^{\mathrm{2}} +{a}=\mathrm{5} \\ $$$${Let}\:{u}={a}+\mathrm{3}\Rightarrow{a}={u}−\mathrm{3} \\ $$$$\Rightarrow\left({u}−\mathrm{3}\right)^{\mathrm{2}} +{u}−\mathrm{3}=\mathrm{5} \\ $$$${u}^{\mathrm{2}} −\mathrm{6}{u}+\mathrm{9}+{u}−\mathrm{3}=\mathrm{5} \\ $$$${u}^{\mathrm{2}} −\mathrm{5}{u}+\mathrm{6}=\mathrm{5} \\ $$$${u}^{\mathrm{2}} −\mathrm{5}{u}+\mathrm{1}=\mathrm{0} \\ $$$${u}=\frac{\mathrm{5}\pm\sqrt{\mathrm{25}−\mathrm{4}×\mathrm{1}}}{\mathrm{2}}=\frac{\mathrm{5}\pm\sqrt{\mathrm{21}}}{\mathrm{2}} \\ $$$${If}\:{u}=\frac{\mathrm{5}+\sqrt{\mathrm{21}}}{\mathrm{2}} \\ $$$${u}^{−\mathrm{1}} =\frac{\mathrm{2}}{\mathrm{5}+\sqrt{\mathrm{21}}}=\frac{\mathrm{2}\left(\mathrm{5}−\sqrt{\mathrm{21}}\right)}{\mathrm{25}−\mathrm{21}}=\frac{\mathrm{5}−\sqrt{\mathrm{21}}}{\mathrm{2}} \\ $$$$\therefore\:{u}^{\mathrm{3}} +{u}^{−\mathrm{3}} =\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{3}} }\left(\left(\mathrm{5}+\sqrt{\mathrm{21}}\right)^{\mathrm{3}} +\left(\mathrm{5}−\sqrt{\mathrm{21}}\right)^{\mathrm{3}} \right) \\ $$$$=\frac{\mathrm{1}}{\mathrm{8}}\left(\mathrm{5}+\sqrt{\mathrm{21}}+\mathrm{5}−\sqrt{\mathrm{21}}\right)\left(\mathrm{5}^{\mathrm{2}} +\mathrm{21}+\mathrm{10}\sqrt{\mathrm{21}}−\mathrm{10}\sqrt{\mathrm{21}}+\mathrm{5}^{\mathrm{2}} +\mathrm{21}+−\mathrm{25}+\mathrm{21}\right) \\ $$$$=\frac{\mathrm{5}}{\mathrm{4}}\left(\mathrm{50}+\mathrm{42}−\mathrm{4}\right) \\ $$$$=\frac{\mathrm{5}}{\mathrm{4}}\left(\mathrm{88}\right) \\ $$$$=\mathrm{5}×\mathrm{22} \\ $$$$=\mathrm{110} \\ $$$$\left({a}+\mathrm{3}\right)^{\mathrm{3}} +\left({a}+\mathrm{3}\right)^{−\mathrm{3}} =\mathrm{110} \\ $$$${If}\:{u}=\frac{\mathrm{5}−\sqrt{\mathrm{21}}}{\mathrm{2}}\Rightarrow{u}^{−\mathrm{1}} =\frac{\mathrm{5}+\sqrt{\mathrm{21}}}{\mathrm{2}} \\ $$$$\Rightarrow{u}^{\mathrm{3}} +{u}^{−\mathrm{3}} =\mathrm{110} \\ $$$${or}\:\left({a}+\mathrm{3}\right)^{\mathrm{3}} +\left({a}+\mathrm{3}\right)^{−\mathrm{3}} =\mathrm{110} \\ $$

Commented by Tawakalitu. last updated on 09/Aug/16

I love the approach, God bless you.

$${I}\:{love}\:{the}\:{approach},\:{God}\:{bless}\:{you}. \\ $$

Answered by Yozzii last updated on 09/Aug/16

Check for an answer in comments.

$${Check}\:{for}\:{an}\:{answer}\:{in}\:{comments}. \\ $$

Commented by Tawakalitu. last updated on 09/Aug/16

Alright sir

$${Alright}\:{sir} \\ $$

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