Question Number 5639 by Daily last updated on 23/May/16
$$\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{negative}\:\mathrm{real}\:\mathrm{roots}\: \\ $$$$\mathrm{of}\:\:\:{x}^{\mathrm{4}} −\mathrm{4}{x}−\mathrm{1}=\mathrm{0}\:\mathrm{is} \\ $$
Commented by Yozzii last updated on 23/May/16
$${Descartes}\:{rule}\:{of}\:{signs}\:{may}\:{help}. \\ $$
Answered by Rasheed Soomro last updated on 27/May/16
$${The}\:{equation}\:{has}\:{no}\:{sign}\:{variation} \\ $$$${so}\:{it}\:{has}\:{no}\:{positive}\:{root}. \\ $$$${Replace}\:{x}\:{by}\:−{x} \\ $$$$\:\:\:\left(−{x}\right)^{\mathrm{4}} −\mathrm{4}\left(−{x}\right)−\mathrm{1}=\mathrm{0} \\ $$$$\Rightarrow{x}^{\mathrm{4}} +\mathrm{4}{x}−\mathrm{1}=\mathrm{0} \\ $$$${This}\:{has}\:\:{one}\:{variation}\:{of}\:{signs} \\ $$$${So}\:{the}\:{given}\:{equation}\:{has}\:{one}\:{negative} \\ $$$${root}. \\ $$