[tex] \sqrt[3]{2-x}+ \sqrt{x+7} =3 \\ \\ $Notam: u = \sqrt[3]{2-x}, \quad v = \sqrt{x+7} \\ \\ $Avem: \left\{ \begin{array}{c} u^3+v^2 = 2-x+x+7= 9 \\ u+v = 3 \end{array} \right \Rightarrow \left\{ \begin{array}{c} u^3+v^2= 9 \\ u+v = 3 \end{array} \right | \\ \\ $Este important sa stim ca \sqrt{x+7} \geq 0$ , adica $ v \geq 0. \\ $Incercam sa ghicim solutiile: \\ \\
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[tex]\boxed{1}$ \quad Daca u = 2, v = 1 \Rightarrow \left\{ \begin{array}{c} 9+1 = 9 \\ 2+1 = 3 \end{array} \right | (A)\Rightarrow$\sqrt{x+7} = 1 \Rightarrow \\ \\ \Rightarrow \boxed{x = -6} \\ \\ \boxed{2} \quad $Daca u = 0, v = 3 \Rightarrow \left\{ \begin{array}{c} 0+9 = 9 \\ 0+3 = 3 \end{array} \right | (A) \Rightarrow \sqrt{x+7} = 3 \Rightarrow \\ \\ \Rightarrow \boxed{x = 2} [/tex]
[tex]\boxed{3} \quad $Daca u = -3, v = 6 \Rightarrow \left\{ \begin{array}{c} -27+36 = 9 \\ -3+6 = 3 \end{array} \right | (A) \Rightarrow \sqrt{x+7} = 6 \Rightarrow \\ \\ \Rightarrow \boxed{x = 29} \\ \\ $Nu stiu explicatia, dar, acestea sunt singurele solutii. \\ \\ \\ \Rightarrow \boxed{\boxed{S = \Big\{-6;2;29\Big\}}}[/tex]
