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2x-1-x-2-x-3-x-4-x-5-13-6-solved-equation-x-lt-1-




Question Number 57164 by ANTARES VY last updated on 31/Mar/19
2x+1+x^2 −x^3 +x^4 −x^5 −.....=((13)/6)  solved  equation.  ∣x∣<1
$$\mathrm{2}\boldsymbol{\mathrm{x}}+\mathrm{1}+\boldsymbol{\mathrm{x}}^{\mathrm{2}} −\boldsymbol{\mathrm{x}}^{\mathrm{3}} +\boldsymbol{\mathrm{x}}^{\mathrm{4}} −\boldsymbol{\mathrm{x}}^{\mathrm{5}} −…..=\frac{\mathrm{13}}{\mathrm{6}} \\ $$$$\boldsymbol{\mathrm{solved}}\:\:\boldsymbol{\mathrm{equation}}. \\ $$$$\mid\boldsymbol{\mathrm{x}}\mid<\mathrm{1} \\ $$
Answered by tanmay.chaudhury50@gmail.com last updated on 31/Mar/19
2x+1+(x^2 /(1+x))=((13)/6)  ((2x+2x^2 +1+x+x^2 )/(1+x))=((13)/6)  18x^2 +18x+6=13+13x  18x^2 +5x−7=0  x=((−5±(√(25+4×18×7)))/(2×18))  =((−5±(√(529)))/(36))  =((−5±23)/(36))→(((18)/(36)),((−28)/(36)))  x=((1/2),((−7)/9))
$$\mathrm{2}{x}+\mathrm{1}+\frac{{x}^{\mathrm{2}} }{\mathrm{1}+{x}}=\frac{\mathrm{13}}{\mathrm{6}} \\ $$$$\frac{\mathrm{2}{x}+\mathrm{2}{x}^{\mathrm{2}} +\mathrm{1}+{x}+{x}^{\mathrm{2}} }{\mathrm{1}+{x}}=\frac{\mathrm{13}}{\mathrm{6}} \\ $$$$\mathrm{18}{x}^{\mathrm{2}} +\mathrm{18}{x}+\mathrm{6}=\mathrm{13}+\mathrm{13}{x} \\ $$$$\mathrm{18}{x}^{\mathrm{2}} +\mathrm{5}{x}−\mathrm{7}=\mathrm{0} \\ $$$${x}=\frac{−\mathrm{5}\pm\sqrt{\mathrm{25}+\mathrm{4}×\mathrm{18}×\mathrm{7}}}{\mathrm{2}×\mathrm{18}} \\ $$$$=\frac{−\mathrm{5}\pm\sqrt{\mathrm{529}}}{\mathrm{36}} \\ $$$$=\frac{−\mathrm{5}\pm\mathrm{23}}{\mathrm{36}}\rightarrow\left(\frac{\mathrm{18}}{\mathrm{36}},\frac{−\mathrm{28}}{\mathrm{36}}\right) \\ $$$${x}=\left(\frac{\mathrm{1}}{\mathrm{2}},\frac{−\mathrm{7}}{\mathrm{9}}\right) \\ $$

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