Question

aratati ca $x_{0}$ = $\sqrt[3]{2}$ +$\sqrt[3]{3}$ verifica ec:$x^{3}$ -6x-6=0

Answer 1

Facem subtitutia $x=y+ \frac{2}{y}$. Ecuatia se scrie:
[tex](y+ \frac{2}{y} )^3-6(y+ \frac{2}{y} )-6=0\\ y^6-6y^3+8=0\\ y^3=z\\ z^2-6z+8=0\\ z= 4 sau 2\\ y^3=4=>y=\sqrt[3]{4}=>x=\sqrt[3]{4}+ \frac{2}{\sqrt[3]{4}} =\sqrt[3]{4}+\frac{\sqrt[3]{8}}{\sqrt[3]{4}}\\ x=\sqrt[3]{4}+\sqrt[3]{2}[/tex]