Question

Sa se determine a apartine R pt care (a-3)x^2-ax-a<0 oricare ar fi x apartine R

Answer 1

$(a-3)x^2-ax-a\ \textless \ 0 \\ \\ Conditii: \\ \\ \boxed{1} \quad a-3\ \textless \ 0 \Rightarrow a\ \textless \ 3 \\ ~\quad \quad \quad \quad \quad \quad si \\ \boxed{2} \quad \Delta \ \textless \ 0 \Rightarrow a^2-4\cdot (a-3)\cdot (-a)\ \textless \ 0 \Rightarrow a^2+4a^2-12a \ \textless \ 0 \Rightarrow \\ \\ \Rightarrow 5a^2-12a \ \textless \ 0 \Big| \quad 5a^2-12a = 0 \Rightarrow a(5a-12) = 0 \Rightarrow \\ \\ \Rightarrow a_1 = 0, a_2 = \dfrac{12}{5} \\ (In tabelul de semn: --- intre radacini, ++++ inafara)\\ \\ \Rightarrow a\in \Big(0,\dfrac{12}{5}\Big)$

$\\ Din \boxed{1} \cap \boxed{2} \Rightarrow \boxed{a \in \Big(0,\dfrac{12}{5}\Big)} \rightarrow solutie ~ finala.$