Ecuația [tex]3x^2-5x-2 = 0[/tex] are:
- [tex]\Delta = (-5)^2-4\cdot 3\cdot (-2) = 25+24 = 49[/tex]
- [tex]x_{1,2} = \frac{5\pm \sqrt{49}}{2\cdot 3} = \frac{5\pm 7}{6}\Rightarrow \left|\begin{aligned}&x_1 = \frac{5-7}{6} = -\frac{1}{3}\\&x_2 =\frac{5+7}{6}=2 \end{aligned}\right.[/tex]
Astfel,
[tex]A = \left\{-\frac{1}{3}; 2\right\}[/tex]
Rezolvarea cerințelor:
[tex]a)\,\,\,A[/tex] / [tex]\mathbb{Z}= \left\{-\frac{1}{3}; 2\right\}[/tex] / [tex]\mathbb{Z} = \left\{-\frac{1}{3}\right\}[/tex]
[tex]b)\,\,\,\text{card}(A\cap \mathbb{N}) = \text{card}\left(\left\{-\frac{1}{3}; 2\right\}\cap \mathbb{N}\right) = \text{card}\{2\} = 1[/tex]
[tex]c)\,\,\,A\cup \left\{-2; 2\right\} = \left\{-\frac{1}{3}; 2\right\}\cup \left\{-2; 2\right\} = \left\{-2;-\frac{1}{3}; 2\right\}[/tex]
[tex]d)\,\,\,A \cap \left[-1; -\frac{1}{2}\right] = \left\{-\frac{1}{3}; 2\right\} \cap \left[-1; -\frac{1}{2}\right] = \varnothing[/tex]
