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Question Number 47581 by EagleEye1 last updated on 11/Nov/18

The root of the equation (2/(x−1)) + (1/(x+2)) + ((3x(x+1))/((x−1)(x+2))) = 0 among the following is

$$\mathrm{The}\:\mathrm{root}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation}\: \\ $$$$\frac{\mathrm{2}}{{x}−\mathrm{1}}\:+\:\frac{\mathrm{1}}{{x}+\mathrm{2}}\:+\:\frac{\mathrm{3}{x}\left({x}+\mathrm{1}\right)}{\left({x}−\mathrm{1}\right)\left({x}+\mathrm{2}\right)}\:=\:\mathrm{0}\:\: \\ $$$$\mathrm{among}\:\mathrm{the}\:\mathrm{following}\:\mathrm{is} \\ $$

Commented by maxmathsup by imad last updated on 12/Nov/18

equation exist if x≠1 and x≠−2 . (e) ⇔ ((2x+4+x−1)/((x−1)(x+2))) +((3x^2 +3x)/((x−1)(x+2))) =0 ⇔((3x+3 +3x^2 +3x)/((x−1)(x+2))) =0 ⇔ 3x^2 +6x +3 =0 ⇔3(x^2 +2x+1)=0 ⇔(x+1)^2 =0 ⇔x=−1 .

$${equation}\:{exist}\:{if}\:\:{x}\neq\mathrm{1}\:{and}\:{x}\neq−\mathrm{2}\:. \\ $$$$\left({e}\right)\:\Leftrightarrow\:\frac{\mathrm{2}{x}+\mathrm{4}+{x}−\mathrm{1}}{\left({x}−\mathrm{1}\right)\left({x}+\mathrm{2}\right)}\:+\frac{\mathrm{3}{x}^{\mathrm{2}} \:+\mathrm{3}{x}}{\left({x}−\mathrm{1}\right)\left({x}+\mathrm{2}\right)}\:=\mathrm{0}\:\Leftrightarrow\frac{\mathrm{3}{x}+\mathrm{3}\:+\mathrm{3}{x}^{\mathrm{2}} \:+\mathrm{3}{x}}{\left({x}−\mathrm{1}\right)\left({x}+\mathrm{2}\right)}\:=\mathrm{0}\:\Leftrightarrow \\ $$$$\mathrm{3}{x}^{\mathrm{2}} \:+\mathrm{6}{x}\:+\mathrm{3}\:=\mathrm{0}\:\Leftrightarrow\mathrm{3}\left({x}^{\mathrm{2}} \:+\mathrm{2}{x}+\mathrm{1}\right)=\mathrm{0}\:\Leftrightarrow\left({x}+\mathrm{1}\right)^{\mathrm{2}} =\mathrm{0}\:\Leftrightarrow{x}=−\mathrm{1}\:. \\ $$

Answered by Rio Michael last updated on 11/Nov/18

((2(x+2)+1(x−1))/(x^2 +x−2))+((3x^2 +3)/(x^2 +x−2))=0 ((3x−3+3x^2 +3)/(x^2 +x−2))=0 ((3x^2 +3x)/(x^2 +x−2))=0 x=0 or x=−1 or x=1 or x=−2

$$\frac{\mathrm{2}\left({x}+\mathrm{2}\right)+\mathrm{1}\left({x}−\mathrm{1}\right)}{{x}^{\mathrm{2}} +{x}−\mathrm{2}}+\frac{\mathrm{3}{x}^{\mathrm{2}} +\mathrm{3}}{{x}^{\mathrm{2}} +{x}−\mathrm{2}}=\mathrm{0} \\ $$$$\:\frac{\mathrm{3}{x}−\mathrm{3}+\mathrm{3}{x}^{\mathrm{2}} +\mathrm{3}}{{x}^{\mathrm{2}} +{x}−\mathrm{2}}=\mathrm{0} \\ $$$$\frac{\mathrm{3}{x}^{\mathrm{2}} +\mathrm{3}{x}}{{x}^{\mathrm{2}} +{x}−\mathrm{2}}=\mathrm{0} \\ $$$${x}=\mathrm{0}\:{or}\:{x}=−\mathrm{1}\:{or}\:{x}=\mathrm{1}\:{or}\:{x}=−\mathrm{2} \\ $$

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