# Exercise ID260

Algebra → Irrational Equations → Köbgyököt tartalmazó egyenletek
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Solve the following equation.

$\sqrt[3]{x}+\sqrt[3]{x+1}+\sqrt[3]{x+2}=0$

$\sqrt[3]{x}+\sqrt[3]{x+1}=-\sqrt[3]{x+2}\phantom{\rule{0ex}{0ex}}$ ${\left(\sqrt[3]{x}+\sqrt[3]{x+1}\right)}^{3}={\left(-\sqrt[3]{x+2}\right)}^{3}\phantom{\rule{0ex}{0ex}}$ ${\left(\sqrt[3]{x}\right)}^{3}+3{\left(\sqrt[3]{x}\right)}^{2}·\sqrt[3]{x+1}+3\sqrt[3]{x}·{\left(\sqrt[3]{x+1}\right)}^{2}+{\left(\sqrt[3]{x+1}\right)}^{3}=-\left(x+2\right)\phantom{\rule{0ex}{0ex}}$ $x+3\sqrt[3]{x}·\sqrt[3]{x+1}\left(\sqrt[3]{x}+\sqrt[3]{x+1}\right)+x+1=-x-2\phantom{\rule{0ex}{0ex}}$ $\sqrt[3]{x}+\sqrt[3]{x+1}+\sqrt[3]{x+2}=0⇒\sqrt[3]{x}+\sqrt[3]{x+1}=-\sqrt[3]{x+2}\phantom{\rule{0ex}{0ex}}$ $x+3\sqrt[3]{x}·\sqrt[3]{x+1}·\sqrt[3]{x+2}\left(-\sqrt[3]{x+2}\right)+x+1=-x-2\phantom{\rule{0ex}{0ex}}$ $-3\sqrt[3]{x}·\sqrt[3]{x+1}·\sqrt[3]{x+2}=-x-x-x-2-1\phantom{\rule{0ex}{0ex}}$ $-3\sqrt[3]{x\left(x+1\right)\left(x+2\right)}=-3x-3\phantom{\rule{0ex}{0ex}}$ $-3\sqrt[3]{x\left(x+1\right)\left(x+2\right)}=-3\left(x+1\right)/·\left(-\frac{1}{3}\right)\phantom{\rule{0ex}{0ex}}$ $\sqrt[3]{x\left(x+1\right)\left(x+2\right)}=x+1\phantom{\rule{0ex}{0ex}}$ ${\left(\sqrt[3]{x\left(x+1\right)\left(x+2\right)}\right)}^{3}={\left(x+1\right)}^{3}\phantom{\rule{0ex}{0ex}}$ $x\left(x+1\right)\left(x+2\right)={\left(x+1\right)}^{3}\phantom{\rule{0ex}{0ex}}$ $x=-1\phantom{\rule{0ex}{0ex}}$
$x=-1$