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Question Number 147419 by Kunal12588 last updated on 20/Jul/21

How can we apply Cardano′s method in 2x^3 +5x^2 +x+2 i get u and v are solution of t^2 −56t+6859=0 but i think it′s wrong pls help

$${How}\:{can}\:{we}\:{apply}\:{Cardano}'{s}\:{method}\:{in} \\ $$$$\mathrm{2}{x}^{\mathrm{3}} +\mathrm{5}{x}^{\mathrm{2}} +{x}+\mathrm{2} \\ $$$$ \\ $$$${i}\:{get}\:{u}\:{and}\:{v}\:{are}\:{solution}\:{of}\:{t}^{\mathrm{2}} −\mathrm{56}{t}+\mathrm{6859}=\mathrm{0} \\ $$$${but}\:{i}\:{think}\:{it}'{s}\:{wrong}\:{pls}\:{help} \\ $$

Answered by mr W last updated on 20/Jul/21

x^3 +((5x^2 )/2)+(x/2)+1=0 let x=s−(5/6) s^3 −3×(5/6)s^2 +3×(5^2 /6^2 )s−(5^3 /6^3 )+(5/2)s^2 −(5/2)×2×(5/6)s+(5/2)×(5^2 /6^2 )+(s/2)−(5/(2×6))+1=0 s^3 −((19)/(12))s+((47)/(27))=0 (√Δ)=(√((−((19)/(36)))^3 +(((47)/(54)))^2 ))=((√(3165))/(72)) s=((((√(3165))/(72))−((47)/(54))))^(1/3) −((((√(3165))/(72))+((47)/(54))))^(1/3) ⇒x=((((√(3165))/(72))−((47)/(54))))^(1/3) −((((√(3165))/(72))+((47)/(54))))^(1/3) −(5/6) ≈−2.416896

$${x}^{\mathrm{3}} +\frac{\mathrm{5}{x}^{\mathrm{2}} }{\mathrm{2}}+\frac{{x}}{\mathrm{2}}+\mathrm{1}=\mathrm{0} \\ $$$${let}\:{x}={s}−\frac{\mathrm{5}}{\mathrm{6}} \\ $$$${s}^{\mathrm{3}} −\mathrm{3}×\frac{\mathrm{5}}{\mathrm{6}}{s}^{\mathrm{2}} +\mathrm{3}×\frac{\mathrm{5}^{\mathrm{2}} }{\mathrm{6}^{\mathrm{2}} }{s}−\frac{\mathrm{5}^{\mathrm{3}} }{\mathrm{6}^{\mathrm{3}} }+\frac{\mathrm{5}}{\mathrm{2}}{s}^{\mathrm{2}} −\frac{\mathrm{5}}{\mathrm{2}}×\mathrm{2}×\frac{\mathrm{5}}{\mathrm{6}}{s}+\frac{\mathrm{5}}{\mathrm{2}}×\frac{\mathrm{5}^{\mathrm{2}} }{\mathrm{6}^{\mathrm{2}} }+\frac{{s}}{\mathrm{2}}−\frac{\mathrm{5}}{\mathrm{2}×\mathrm{6}}+\mathrm{1}=\mathrm{0} \\ $$$${s}^{\mathrm{3}} −\frac{\mathrm{19}}{\mathrm{12}}{s}+\frac{\mathrm{47}}{\mathrm{27}}=\mathrm{0} \\ $$$$\sqrt{\Delta}=\sqrt{\left(−\frac{\mathrm{19}}{\mathrm{36}}\right)^{\mathrm{3}} +\left(\frac{\mathrm{47}}{\mathrm{54}}\right)^{\mathrm{2}} }=\frac{\sqrt{\mathrm{3165}}}{\mathrm{72}} \\ $$$${s}=\sqrt[{\mathrm{3}}]{\frac{\sqrt{\mathrm{3165}}}{\mathrm{72}}−\frac{\mathrm{47}}{\mathrm{54}}}−\sqrt[{\mathrm{3}}]{\frac{\sqrt{\mathrm{3165}}}{\mathrm{72}}+\frac{\mathrm{47}}{\mathrm{54}}} \\ $$$$\Rightarrow{x}=\sqrt[{\mathrm{3}}]{\frac{\sqrt{\mathrm{3165}}}{\mathrm{72}}−\frac{\mathrm{47}}{\mathrm{54}}}−\sqrt[{\mathrm{3}}]{\frac{\sqrt{\mathrm{3165}}}{\mathrm{72}}+\frac{\mathrm{47}}{\mathrm{54}}}−\frac{\mathrm{5}}{\mathrm{6}} \\ $$$$\approx−\mathrm{2}.\mathrm{416896} \\ $$

Commented by Kunal12588 last updated on 21/Jul/21

thank you sir

$${thank}\:{you}\:{sir} \\ $$

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