Question

Bună ziua! Cum as putea sa rezolv problema 268?

Answer 1

Răspuns:

Explicație pas cu pas:

$\texttt{Fie }\alpha\in\mathbb{R}\texttt{ radacina dubla comuna.Atunci:}\\f(\alpha)=f'(\alpha)=g(\alpha)=g'(\alpha), \texttt{ unde }f=x^4+4x^3+4x^2+a\\\texttt{ si } g=x^3+x^2-x+b\\\texttt{Din }f'(\alpha)=0\texttt{ obtinem }4\alpha^3+12\alpha^2+8\alpha=0,\texttt{adica }4\alpha(\alpha+1)(\alpha+2)=0\\ \texttt{Pe de alta parte , din }g'(\alpha)=0\texttt{ obtinem }3\alpha^2+2\alpha-1=0,\texttt{de unde}\\(\alpha+1)(3\alpha-1)=0 . \texttt{Observam ca }\alpha=-1\texttt{ este solutie comuna pt.}$

$\texttt{ambele ecuatii, deci e singura solutie convenabila. }\\f(-1)=0\Rightarrow a=-1\\g(-1)=0\Rightarrow b=-1\\\texttt{Raspunsul corect este a)}.$