Question Number 138391 by ajfour last updated on 13/Apr/21
$${x}^{\mathrm{3}} −{x}−{c}=\mathrm{0}\:\:;\:\mathrm{0}<{c}<\frac{\mathrm{2}}{\mathrm{3}\sqrt{\mathrm{3}}} \\ $$$${Find}\:{x}. \\ $$
Answered by ajfour last updated on 13/Apr/21
$${let}\:\:\:{t}^{\mathrm{2}} +{pt}+{q}=\mathrm{0} \\ $$$$\left({x}^{\mathrm{3}} −{t}^{\mathrm{3}} \right)−{pt}^{\mathrm{2}} −\left({x}+{qt}\right)−{c}=\mathrm{0} \\ $$$${let}\:\:{x}−{t}={z} \\ $$$$\left({x}−{t}\right)\left\{\left({x}−{t}\right)^{\mathrm{2}} +\mathrm{3}{tx}\right\}−{pt}^{\mathrm{2}} −\left({x}−{t}\right) \\ $$$$+\left({q}+\mathrm{1}\right){t}−{c}=\mathrm{0} \\ $$$${z}\left({z}^{\mathrm{2}} +\mathrm{3}{tz}+\mathrm{3}{t}^{\mathrm{2}} \right)−{z}−\left\{{pt}^{\mathrm{2}} −\left({q}+\mathrm{1}\right){t}+{c}\right\}=\mathrm{0} \\ $$$${z}^{\mathrm{3}} +\mathrm{3}{tz}^{\mathrm{2}} +\left(\mathrm{3}{t}^{\mathrm{2}} −\mathrm{1}\right){z}={pt}^{\mathrm{2}} −\left({q}+\mathrm{1}\right){t}+{c} \\ $$$${let}\:\:{q}=−\mathrm{1} \\ $$$${pt}^{\mathrm{2}} =−{c} \\ $$$$\&\:\:{t}^{\mathrm{2}} +{pt}−\mathrm{1}=\mathrm{0} \\ $$$$−\frac{{c}}{{p}}+{pt}−\mathrm{1}=\mathrm{0} \\ $$$${t}=\frac{{p}+{c}}{{p}^{\mathrm{2}} } \\ $$$${p}\left(\frac{{p}+{c}}{{p}^{\mathrm{2}} }\right)^{\mathrm{2}} =−{c}\:\:\:\:\:\:\left[{p}<\mathrm{0}\right] \\ $$$${cp}^{\mathrm{3}} +{p}^{\mathrm{2}} +\mathrm{2}{cp}+{c}^{\mathrm{2}} =\mathrm{0} \\ $$$${p}^{\mathrm{3}} +\frac{{p}^{\mathrm{2}} }{{c}}+\mathrm{2}{p}+{c}=\mathrm{0} \\ $$$${let}\:\:{p}={s}−\frac{\mathrm{1}}{\mathrm{3}{c}} \\ $$$${s}^{\mathrm{3}} −\frac{{s}^{\mathrm{2}} }{{c}}+\frac{{s}}{\mathrm{3}{c}^{\mathrm{2}} }−\frac{\mathrm{1}}{\mathrm{27}{c}^{\mathrm{3}} } \\ $$$$+\frac{\mathrm{1}}{{c}}\left({s}^{\mathrm{2}} −\frac{\mathrm{2}{s}}{\mathrm{3}{c}}+\frac{\mathrm{1}}{\mathrm{9}{c}^{\mathrm{2}} }\right)+\mathrm{2}{s}−\frac{\mathrm{2}}{\mathrm{3}{c}}+{c}=\mathrm{0} \\ $$$$\Rightarrow\:\:{s}^{\mathrm{3}} +\left(\mathrm{2}−\frac{\mathrm{1}}{\mathrm{3}{c}^{\mathrm{2}} }\right){s}+\frac{\mathrm{2}}{\mathrm{27}{c}^{\mathrm{3}} }−\frac{\mathrm{2}}{\mathrm{3}{c}}+{c}=\mathrm{0} \\ $$$${D}=\left(\frac{\mathrm{1}}{\mathrm{27}{c}^{\mathrm{3}} }−\frac{\mathrm{1}}{\mathrm{3}{c}}+\frac{{c}}{\mathrm{2}}\right)^{\mathrm{2}} −\left(\frac{\mathrm{1}}{\mathrm{9}{c}^{\mathrm{2}} }−\frac{\mathrm{2}}{\mathrm{3}}\right)^{\mathrm{3}} \\ $$$$\:\:=\frac{\mathrm{1}}{\mathrm{9}{c}^{\mathrm{2}} }+\frac{{c}^{\mathrm{2}} }{\mathrm{4}}−\frac{\mathrm{2}}{\mathrm{81}{c}^{\mathrm{4}} }−\frac{\mathrm{1}}{\mathrm{3}}+\frac{\mathrm{1}}{\mathrm{27}{c}^{\mathrm{2}} } \\ $$$$\:\:+\frac{\mathrm{8}}{\mathrm{27}}+\frac{\mathrm{2}}{\mathrm{81}{c}^{\mathrm{4}} }−\frac{\mathrm{4}}{\mathrm{27}{c}^{\mathrm{2}} } \\ $$$${D}=\frac{{c}^{\mathrm{2}} }{\mathrm{4}}−\frac{\mathrm{1}}{\mathrm{27}} \\ $$$$\left({same}\:{old}\:{tragedy}..\right) \\ $$