Question

............................

Answer 1

Răspuns:

Explicație pas cu pas:

$E=\sqrt[3]{10+6\sqrt 3}+\sqrt[3]{10-6\sqrt3}\\E^3=(\sqrt[3]{10+6\sqrt 3}+\sqrt[3]{10-6\sqrt3})^3\\E^3=10+6\sqrt3+10-6\sqrt 3+3\sqrt[3]{(10+6\sqrt 3)(10-6\sqrt 3)}\cdot E\\E^3=20+3\sqrt[3]{100-108}\cdot E\\E^3=20+3\sqrt[3]{-8}\cdot E\\E^3=20-3\cdot 2\cdotE\\E^3=20-6E\\E^3+6E-20=0\\E^3-2E^2+2E^2-4E+10E-20=0\\E^2(E-2)+2E(E-2)+10(E-2)=0\\(E-2)(E^2+2E+10)=0\\i)E-2=0\Rightarrow E=2\\ii)E^2+2E+10=0\\\Delta=4-4\cdot 10=-36<0\\\texttt{Prin urmare }\boxed{E=2}$

$\texttt{Daca E este solutie a ecuatiei, atunci este clar ca}\\E^3+aE+b=0\\8+2a+b=0\\2a+b=-8\\\texttt{Raspunsul corect este }d)$