Question Number 147941 by mathdanisur last updated on 24/Jul/21
$${x}^{\mathrm{2}} \:-\:\mathrm{6}{x}\:+\:\mathrm{3}\:=\:\mathrm{0}\:\:\:\Rightarrow\:\:\:\frac{\mathrm{2}{x}^{\mathrm{2}} }{\mathrm{3}}\:+\:\frac{\mathrm{6}}{{x}^{\mathrm{2}} }\:=\:? \\ $$
Answered by Rasheed.Sindhi last updated on 24/Jul/21
$${x}^{\mathrm{2}} \:-\:\mathrm{6}{x}\:+\:\mathrm{3}\:=\:\mathrm{0}\Rightarrow{x}^{\mathrm{2}} =\mathrm{6}{x}−\mathrm{3} \\ $$$$\frac{\mathrm{2}{x}^{\mathrm{2}} }{\mathrm{3}}\:+\:\frac{\mathrm{6}}{{x}^{\mathrm{2}} }\:=\frac{\mathrm{2}\left(\mathrm{6}{x}−\mathrm{3}\right)}{\mathrm{3}}+\frac{\mathrm{6}}{\mathrm{6}{x}−\mathrm{3}} \\ $$$$=\mathrm{4}{x}−\mathrm{2}+\frac{\mathrm{2}}{\mathrm{2}{x}−\mathrm{1}}=\mathrm{2}\left(\mathrm{2}{x}−\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}{x}−\mathrm{1}}\right) \\ $$$${x}^{\mathrm{2}} \:-\:\mathrm{6}{x}\:+\:\mathrm{3}\:=\:\mathrm{0}\Rightarrow{x}=\frac{\mathrm{6}\pm\sqrt{\mathrm{36}−\mathrm{12}}}{\mathrm{2}} \\ $$$$\:\:\:{x}=\mathrm{3}\pm\sqrt{\mathrm{6}}\Rightarrow\mathrm{2}{x}−\mathrm{1}=\mathrm{5}\pm\mathrm{2}\sqrt{\mathrm{6}} \\ $$$$\frac{\mathrm{2}{x}^{\mathrm{2}} }{\mathrm{3}}\:+\:\frac{\mathrm{6}}{{x}^{\mathrm{2}} }\:=\mathrm{2}\left(\mathrm{2}{x}−\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}{x}−\mathrm{1}}\right) \\ $$$$=\mathrm{2}\left(\mathrm{5}\pm\mathrm{2}\sqrt{\mathrm{6}}+\frac{\mathrm{1}}{\mathrm{5}\pm\mathrm{2}\sqrt{\mathrm{6}}}\right) \\ $$$$\:\frac{\mathrm{1}}{\mathrm{5}\pm\mathrm{2}\sqrt{\mathrm{6}}}=\frac{\mathrm{1}}{\mathrm{5}\pm\mathrm{2}\sqrt{\mathrm{6}}}×\frac{\mathrm{5}\mp\mathrm{2}\sqrt{\mathrm{6}}}{\mathrm{5}\mp\mathrm{2}\sqrt{\mathrm{6}}}=\frac{\mathrm{5}\mp\mathrm{2}\sqrt{\mathrm{6}}}{\mathrm{25}−\mathrm{24}}=\mathrm{5}\mp\mathrm{2}\sqrt{\mathrm{6}} \\ $$$$\mathrm{2}\left(\mathrm{5}\pm\cancel{\mathrm{2}\sqrt{\mathrm{6}}}+\mathrm{5}\mp\cancel{\mathrm{2}\sqrt{\mathrm{6}}}\right)=\mathrm{20} \\ $$
Commented by mathdanisur last updated on 24/Jul/21
$${Thank}\:{you}\:{Ser} \\ $$
Answered by lyubita last updated on 25/Jul/21
$${x}^{\mathrm{2}} \:+\:\mathrm{3}\:=\:\mathrm{6}{x} \\ $$$${x}\:+\:\frac{\mathrm{3}}{{x}}\:=\:\mathrm{6} \\ $$$${x}^{\mathrm{2}} \:+\:\frac{\mathrm{9}}{{x}^{\mathrm{2}} }\:=\:\mathrm{30} \\ $$$$\frac{\mathrm{2}{x}^{\mathrm{2}} }{\mathrm{3}}\:+\:\frac{\mathrm{6}}{{x}^{\mathrm{2}\:} }\:=\:\mathrm{20} \\ $$
Commented by Rasheed.Sindhi last updated on 25/Jul/21
$$\mathcal{N}{ice}! \\ $$
Commented by lyubita last updated on 25/Jul/21
$${thank}\:{you}\:{mr}\:{rasheed} \\ $$