Question-223858

Question Number 223858 by Rojarani last updated on 07/Aug/25

Answered by Rasheed.Sindhi last updated on 08/Aug/25

let x+100=y (((y−2)^5 +(y+2)^5 )/((y−1)^5 +(y+1)^5 ))=((16^2 )/(5^2 +6^2 )) ((2y^5 +80y^3 +160y)/(2y^5 +20y^3 +10y))=((256)/(61)) ((y^4 +40y^2 +80)/(y^4 +10y^2 +5))=((256)/(61)) 256y^4 +2560y^2 +1280=61y^4 +2440y^2 +4880 195y^4 +120y^2 −3600=0 13y^4 +8y^2 −240=0 (y−2)(y+2)(13y^2 +60)=0 y=2⇒x+100=2⇒x=−98 ✓ y=−2⇒x+100=−2⇒x=−102✓ y^2 =−((60)/(13))⇒(x+100)^2 =−((60)/(13)) ⇒x^2 +200x+10000=−((60)/(13)) 13x^2 +2600x+130060=0 x_1 =−100+((2i(√(195)) )/(13)) ✓ x_2 =−100−((2i(√(195)) )/(13)) ✓

$${let}\:{x}+\mathrm{100}={y} \\ $$$$\frac{\left({y}−\mathrm{2}\right)^{\mathrm{5}} +\left({y}+\mathrm{2}\right)^{\mathrm{5}} }{\left({y}−\mathrm{1}\right)^{\mathrm{5}} +\left({y}+\mathrm{1}\right)^{\mathrm{5}} }=\frac{\mathrm{16}^{\mathrm{2}} }{\mathrm{5}^{\mathrm{2}} +\mathrm{6}^{\mathrm{2}} } \\ $$$$\frac{\mathrm{2}{y}^{\mathrm{5}} +\mathrm{80}{y}^{\mathrm{3}} +\mathrm{160}{y}}{\mathrm{2}{y}^{\mathrm{5}} +\mathrm{20}{y}^{\mathrm{3}} +\mathrm{10}{y}}=\frac{\mathrm{256}}{\mathrm{61}} \\ $$$$\frac{{y}^{\mathrm{4}} +\mathrm{40}{y}^{\mathrm{2}} +\mathrm{80}}{{y}^{\mathrm{4}} +\mathrm{10}{y}^{\mathrm{2}} +\mathrm{5}}=\frac{\mathrm{256}}{\mathrm{61}} \\ $$$$\mathrm{256}{y}^{\mathrm{4}} +\mathrm{2560}{y}^{\mathrm{2}} +\mathrm{1280}=\mathrm{61}{y}^{\mathrm{4}} +\mathrm{2440}{y}^{\mathrm{2}} +\mathrm{4880} \\ $$$$\mathrm{195}{y}^{\mathrm{4}} +\mathrm{120}{y}^{\mathrm{2}} −\mathrm{3600}=\mathrm{0} \\ $$$$\mathrm{13}{y}^{\mathrm{4}} +\mathrm{8}{y}^{\mathrm{2}} −\mathrm{240}=\mathrm{0} \\ $$$$\left({y}−\mathrm{2}\right)\left({y}+\mathrm{2}\right)\left(\mathrm{13}{y}^{\mathrm{2}} +\mathrm{60}\right)=\mathrm{0} \\ $$$${y}=\mathrm{2}\Rightarrow{x}+\mathrm{100}=\mathrm{2}\Rightarrow{x}=−\mathrm{98}\:\checkmark \\ $$$${y}=−\mathrm{2}\Rightarrow{x}+\mathrm{100}=−\mathrm{2}\Rightarrow{x}=−\mathrm{102}\checkmark \\ $$$${y}^{\mathrm{2}} =−\frac{\mathrm{60}}{\mathrm{13}}\Rightarrow\left({x}+\mathrm{100}\right)^{\mathrm{2}} =−\frac{\mathrm{60}}{\mathrm{13}} \\ $$$$\Rightarrow{x}^{\mathrm{2}} +\mathrm{200}{x}+\mathrm{10000}=−\frac{\mathrm{60}}{\mathrm{13}} \\ $$$$\:\:\:\mathrm{13}{x}^{\mathrm{2}} +\mathrm{2600}{x}+\mathrm{130060}=\mathrm{0} \\ $$$${x}_{\mathrm{1}} =−\mathrm{100}+\frac{\mathrm{2}{i}\sqrt{\mathrm{195}}\:}{\mathrm{13}}\:\checkmark \\ $$$${x}_{\mathrm{2}} =−\mathrm{100}−\frac{\mathrm{2}{i}\sqrt{\mathrm{195}}\:}{\mathrm{13}}\:\checkmark \\ $$

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