Developed-the-formula-for-the-cubic-equation-when-it-has-three-real-roots-and-Cardano-is-just-not-suitable-as-discriminant-gets-negative-Given-x-3-bx-c-0-with-b-gt-0-c-gt-0-t-b-1-15

Question Number 114388 by ajfour last updated on 18/Sep/20

$${Developed}\:{the}\:{formula}\:{for}\:{the}\:{cubic} \\ $$$${equation}\:{when}\:{it}\:{has}\:{three}\:{real}\:{roots} \\ $$$${and}\:{Cardano}\:{is}\:{just}\:{not}\:{suitable}\:{as} \\ $$$${discriminant}\:{gets}\:{negative}.. \\ $$$$\:\:\:{Given}\:\:\:\:\boldsymbol{{x}}^{\mathrm{3}} −\boldsymbol{{bx}}−\boldsymbol{{c}}=\mathrm{0} \\ $$$$\:\:{with}\:\:\boldsymbol{{b}}>\mathrm{0}\:,\:\boldsymbol{{c}}>\mathrm{0} \\ $$$$\:\frac{\boldsymbol{{t}}}{\boldsymbol{{b}}}=\mathrm{1}+\frac{\mathrm{15}}{\mathrm{2}}\left(\frac{\boldsymbol{{c}}^{\mathrm{2}} }{\boldsymbol{{b}}^{\mathrm{3}} }\right)+\sqrt{\left(\frac{\boldsymbol{{c}}^{\mathrm{2}} }{\boldsymbol{{b}}^{\mathrm{3}} }\right)\left[\mathrm{49}−\frac{\mathrm{75}}{\mathrm{4}}\left(\frac{\boldsymbol{{c}}^{\mathrm{2}} }{\boldsymbol{{b}}^{\mathrm{3}} }\right)\right]} \\ $$$$\:\:\:\:\:\boldsymbol{{x}}=\sqrt{\boldsymbol{{t}}+\frac{\mathrm{4}\boldsymbol{{c}}^{\mathrm{2}} }{\boldsymbol{{b}}^{\mathrm{2}} }}\:−\frac{\mathrm{3}\boldsymbol{{c}}}{\boldsymbol{{b}}}\:\:\bigstar \\ $$$${mrW}\:{Sir},\:{MjS}\:{Sir}\:\:{please}\:{check}! \\ $$

Commented by MJS_new last updated on 18/Sep/20

$$\mathrm{it}'\mathrm{s}\:\mathrm{not}\:\mathrm{exact} \\ $$$${x}^{\mathrm{3}} −\frac{\mathrm{244}}{\mathrm{3}}{x}−\frac{\mathrm{1456}}{\mathrm{27}}=\mathrm{0}\:\Rightarrow\:{x}\in\left\{−\frac{\mathrm{26}}{\mathrm{3}},\:−\frac{\mathrm{2}}{\mathrm{3}},\:\frac{\mathrm{28}}{\mathrm{3}}\right\} \\ $$$$\mathrm{but}\:\mathrm{your}\:\mathrm{formula}\:\mathrm{gives} \\ $$$${x}\approx\mathrm{9}.\mathrm{33355157}\:\mathrm{instead}\:\mathrm{of}\:\mathrm{9}.\overset{\bullet} {\mathrm{3}} \\ $$

Commented by ajfour last updated on 18/Sep/20

$${yes}\:{sir},\:{i}\:{found}\:{the}\:{error}!\:{thanks}. \\ $$

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