# Exercise ID262

Algebra → Irrational Equations → Köbgyököt tartalmazó egyenletek
[Level: ] [Number of helps: 0] [Number of pictures: 0] [Number of steps: 20] [Number of characters: 0]

Solve the following equation.

$\sqrt[3]{x}+\sqrt[3]{2x-3}=\sqrt[3]{12\left(x-1\right)}$

$\sqrt[3]{x}+\sqrt[3]{2x-3}=\sqrt[3]{12\left(x-1\right)}\phantom{\rule{0ex}{0ex}}$ ${\left(\sqrt[3]{x}+\sqrt[3]{2x-3}\right)}^{3}={\left(\sqrt[3]{12\left(x-1\right)}\right)}^{3}\phantom{\rule{0ex}{0ex}}$ ${\left(\sqrt[3]{x}\right)}^{3}+3{\left(\sqrt[3]{x}\right)}^{2}·\sqrt[3]{2x-3}+3\sqrt[3]{x}{\left(\sqrt[3]{2x-3}\right)}^{2}+{\left(\sqrt[3]{2x-3}\right)}^{3}=12\left(x-1\right)\phantom{\rule{0ex}{0ex}}$ $x+3·\sqrt[3]{x}·\sqrt[3]{2x-3}\left(\sqrt[3]{x}+\sqrt[3]{2x-3}\right)+2x-3=12x-12\phantom{\rule{0ex}{0ex}}$ $x+3·\sqrt[3]{x}·\sqrt[3]{2x-3}·\sqrt[3]{12\left(x-1\right)}+2x-3=12x-12\phantom{\rule{0ex}{0ex}}$ $3\sqrt[3]{x\left(2x-3\right)12\left(x-1\right)}=12x-x-2x-12+3\phantom{\rule{0ex}{0ex}}$ $3\sqrt[3]{12x\left(2x-3\right)\left(x-1\right)}=9x-9\phantom{\rule{0ex}{0ex}}$ ${\left(3\sqrt[3]{12x\left(2x-3\right)\left(x-1\right)}\right)}^{3}={9}^{3}{\left(x-1\right)}^{3}\phantom{\rule{0ex}{0ex}}$ ${3}^{3}·12x\left(2x-3\right)\left(x-1\right)={\left({3}^{2}\right)}^{3}{\left(x-1\right)}^{3}\phantom{\rule{0ex}{0ex}}$ ${3}^{3}·3·4x\left(2x-3\right)\left(x-1\right)-{3}^{6}·{\left(x-1\right)}^{3}=0/·\frac{1}{{3}^{4}}\phantom{\rule{0ex}{0ex}}$ $4x\left(2x-3\right)\left(x-1\right)-{3}^{2}{\left(x-1\right)}^{3}=0\phantom{\rule{0ex}{0ex}}$ $\left(x-1\right)\left[4x\left(2x-3\right)-9{\left(x-1\right)}^{2}\right]=0\phantom{\rule{0ex}{0ex}}$ $\left(x-1\right)\left[8{x}^{2}-12x-9\left({x}^{2}-2x+1\right)\right]=0\phantom{\rule{0ex}{0ex}}$ $\left(x-1\right)\left(8{x}^{2}-12x-9{x}^{2}+18x-9\right)=0\phantom{\rule{0ex}{0ex}}$ $\left(x-1\right)\left(-{x}^{2}+6x-9\right)=0\phantom{\rule{0ex}{0ex}}$ $-{x}^{2}+6x-9=0/·\left(-1\right)\phantom{\rule{0ex}{0ex}}$ ${x}_{1,2}=\frac{6±\sqrt{{\left(-6\right)}^{2}-4·1·9}}{2}=\frac{6±\sqrt{36-36}}{2}=\frac{6±0}{2}\phantom{\rule{0ex}{0ex}}$ ${x}_{1}={x}_{2}=3\phantom{\rule{0ex}{0ex}}$