Question

x1+x2=?
$\sqrt[3]{1 + \sqrt{x} } + \sqrt[3]{8 - \sqrt{x} } = 3$
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Answer 1

Răspuns:

Explicație pas cu pas:

$\texttt{Ridicam ecuatia la puterea a 3-a:}\\(\sqrt[3]{1+\sqrt x}+\sqrt[3]{8-\sqrt x})^3=3^3\\1+\sqrt x+8-\sqrt x+3\sqrt[3]{(1+\sqrt x)(8-\sqrt x)}\underbrace{(\sqrt[3]{1+\sqrt x}+\sqrt[3]{8-\sqrt x})}=27\\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=3\\ 9+3\cdot 3\sqrt[3]{(1+\sqrt x)(8-\sqrt x)}=27\\9+9\cdot \sqrt[3]{(1+\sqrt x)(8-\sqrt x)}=27|:9\\1+\sqrt[3]{(1+\sqrt x)(8-\sqrt x)}=3\\\sqrt[3]{(1+\sqrt x)(8-\sqrt x)}=2|()^3\\(1+\sqrt{x})(8-\sqrt x)=8\\8-\sqrt{x}+8\sqrt{x}-x=8$

$7\sqrt{x}-x=0\\ \sqrt{x}\stackrel{\texttt{not}}{=} t,t\geqslant 0\\7t-t^2=0\\t(7-t)=0\\t=0\Rightarrow x_1=0\\t=7\Rightarrow x_2=49\\\boxed{x_1+x_2=49}$