[tex] \sqrt{2-x}+ \sqrt{x+7 } =3 \\ \\ $Notam: $ \sqrt{2-x} = a; \quad \sqrt{x+7} = b \\ \\ \Big\{a+b =3 \quad $si$ \quad a^2+b^2 = 2-x-x+7 \Big\} \Rightarrow \\ \\ \Rightarrow \Big\{a+b =3 \quad $si$ \quad a^2+b^2=9\Big\} \\ \\ (a+b)^2 =a^2+b^2+2ab \Rightarrow 3^2 = 9+2ab \Rightarrow 9 = 9 +2ab \Rightarrow \\ \Rightarrow 2ab = 0 \Rightarrow ab = 0 \\ \\ $Avem: $\quad \Big\{a+b = 3 \quad $si$ \quad ab = 0 \Big\}[/tex]
[tex]$ \ $Consideram: $ t^2-St+P = 0 \Rightarrow t^2-(a+b)t + ab = 0 \Rightarrow \\ \\ \Rightarrow t^2-3t+ 0 = 0 \Rightarrow t^2 - 3t = 0 \Rightarrow t(t-3) = 0 \Rightarrow t_1 = 0, \quad t_2 = 3 \\ \\ \boxed{1} \quad \Big\{a = 0\quad $si$ \quad b = 3\Big\} \Rightarrow \Big\{\sqrt{2-x} = 0 \quad $si$ \quad \sqrt{x+7} = 3\Big\}\Rightarrow \\ \Rightarrow \Big\{2-x = 0 \quad $si$ \quad x+7 = 9\Big\} \Rightarrow \Big\{x=2 \quad $si$ \quad x = 2\Big\} \Rightarrow \boxed{x = 2} [/tex]
[tex]\boxed{2} \quad \Big\{a = 3 \quad $si$ \quad b = 0\Big\} \Rightarrow \Big\{\sqrt{2-x} = 3 \quad $si$ \quad \sqrt{x+7} = 0\Big\}\Rightarrow \\ \Rightarrow \Big\{2-x = 9 \quad $si$ \quad x+7 = 0\Big\} \Rightarrow \Big\{x=-7 \quad $si$ \quad x = -7\Big\} \Rightarrow \boxed{x=-7} \\ \\ \\ $Din $\boxed{1}$ $\cup $ $ \boxed{2} \Rightarrow x \in \Big\{-7,2\Big\}, \\ $Dar, x trebuie sa apartina numerelor naturale $ \Rightarrow \boxed{S = \big\{2\big\}}[/tex]
