Question Number 147419 by Kunal12588 last updated on 20/Jul/21
$${How}\:{can}\:{we}\:{apply}\:{Cardano}'{s}\:{method}\:{in} \\ $$$$\mathrm{2}{x}^{\mathrm{3}} +\mathrm{5}{x}^{\mathrm{2}} +{x}+\mathrm{2} \\ $$$$ \\ $$$${i}\:{get}\:{u}\:{and}\:{v}\:{are}\:{solution}\:{of}\:{t}^{\mathrm{2}} −\mathrm{56}{t}+\mathrm{6859}=\mathrm{0} \\ $$$${but}\:{i}\:{think}\:{it}'{s}\:{wrong}\:{pls}\:{help} \\ $$
Answered by mr W last updated on 20/Jul/21
$${x}^{\mathrm{3}} +\frac{\mathrm{5}{x}^{\mathrm{2}} }{\mathrm{2}}+\frac{{x}}{\mathrm{2}}+\mathrm{1}=\mathrm{0} \\ $$$${let}\:{x}={s}−\frac{\mathrm{5}}{\mathrm{6}} \\ $$$${s}^{\mathrm{3}} −\mathrm{3}×\frac{\mathrm{5}}{\mathrm{6}}{s}^{\mathrm{2}} +\mathrm{3}×\frac{\mathrm{5}^{\mathrm{2}} }{\mathrm{6}^{\mathrm{2}} }{s}−\frac{\mathrm{5}^{\mathrm{3}} }{\mathrm{6}^{\mathrm{3}} }+\frac{\mathrm{5}}{\mathrm{2}}{s}^{\mathrm{2}} −\frac{\mathrm{5}}{\mathrm{2}}×\mathrm{2}×\frac{\mathrm{5}}{\mathrm{6}}{s}+\frac{\mathrm{5}}{\mathrm{2}}×\frac{\mathrm{5}^{\mathrm{2}} }{\mathrm{6}^{\mathrm{2}} }+\frac{{s}}{\mathrm{2}}−\frac{\mathrm{5}}{\mathrm{2}×\mathrm{6}}+\mathrm{1}=\mathrm{0} \\ $$$${s}^{\mathrm{3}} −\frac{\mathrm{19}}{\mathrm{12}}{s}+\frac{\mathrm{47}}{\mathrm{27}}=\mathrm{0} \\ $$$$\sqrt{\Delta}=\sqrt{\left(−\frac{\mathrm{19}}{\mathrm{36}}\right)^{\mathrm{3}} +\left(\frac{\mathrm{47}}{\mathrm{54}}\right)^{\mathrm{2}} }=\frac{\sqrt{\mathrm{3165}}}{\mathrm{72}} \\ $$$${s}=\sqrt[{\mathrm{3}}]{\frac{\sqrt{\mathrm{3165}}}{\mathrm{72}}−\frac{\mathrm{47}}{\mathrm{54}}}−\sqrt[{\mathrm{3}}]{\frac{\sqrt{\mathrm{3165}}}{\mathrm{72}}+\frac{\mathrm{47}}{\mathrm{54}}} \\ $$$$\Rightarrow{x}=\sqrt[{\mathrm{3}}]{\frac{\sqrt{\mathrm{3165}}}{\mathrm{72}}−\frac{\mathrm{47}}{\mathrm{54}}}−\sqrt[{\mathrm{3}}]{\frac{\sqrt{\mathrm{3165}}}{\mathrm{72}}+\frac{\mathrm{47}}{\mathrm{54}}}−\frac{\mathrm{5}}{\mathrm{6}} \\ $$$$\approx−\mathrm{2}.\mathrm{416896} \\ $$
Commented by Kunal12588 last updated on 21/Jul/21
$${thank}\:{you}\:{sir} \\ $$