Question

Va rog o solutie complete?!

Answer 1

CE x≥-7

∛(2-x) descrescatoare pr R
√(x+7) crescatoare pe dom de definitie
deci ecuatia are cel mult 2 solutii, pt ca poate avea cel mult un minim

se observa ca x=2, care∈Domeniuluide def., verifica
∛0+√9=0+3=3
si ca x=-6, care ∈Dom.de definitie, iarasi verifica
∛8+√1=2+1=3
cum ecuatia are cel mult 2 solutii si am gasit 2 solutii, nu mai cautam altele
deci S={-6;2}

Answer 2

[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: \\ \\ [/tex]

$\boxed{1} \quad Daca u = 2, v = 1 \Rightarrow \left\{ \begin{array}{c} 9+1 = 9 \\ 2+1 = 3 \end{array} \right | (A)\Rightarrow \sqrt{x+7} = 1 \Rightarrow \\ \\ \Rightarrow \boxed{x = -6} \\ \\ \boxed{2} \quad Daca u = 0, v = 3 \Rightarrow \left\{ \begin{array}{c} 0+9 = 9 \\ 0+3 = 3 \end{array} \right | (A) \Rightarrow \sqrt{x+7} = 3 \Rightarrow \\ \\ \Rightarrow \boxed{x = 2}$

$\boxed{3} \quad Daca u = -3, v = 6 \Rightarrow \left\{ \begin{array}{c} -27+36 = 9 \\ -3+6 = 3 \end{array} \right | (A) \Rightarrow \sqrt{x+7} = 6 \Rightarrow \\ \\ \Rightarrow \boxed{x = 29} \\ \\ Nu stiu explicatia, dar, acestea sunt singurele solutii. \\ \\ \\ \Rightarrow \boxed{\boxed{S = \Big\{-6;2;29\Big\}}}$