Question

Solve the equation $\sqrt[3]{3(2x-1)}+\sqrt[3]{3(x-1)}=1$

Answer 1

We have $\sqrt[3]{3(2x-1)}+\sqrt[3]{3(x-1)}=1$............(1)
Cubing both sides of (1) , we obtain
$2x-1+x-1+3\sqrt[3]{3(2x-1)(x-1)}(\sqrt[3]{3(2x-1)}+\sqrt[3]{3(x-1)})=1$
$\Rightarrow 3x-2+3.\sqrt[3]{3({2x}^2-3x+1)}\left(1\right)=1$ {from(1)}
$\Rightarrow 3.\sqrt[3]{3({2x}^2-3x+1)}=3-3x$
$\Rightarrow \sqrt[3]{3({2x}^2-3x+1)}=(1-x)$
again cubing both sides, we obtain
${2x}^2-3x+1={(1-x)}^3\Rightarrow (2x-1)(x-1)={(1-x)}^3\Rightarrow (2x-1)(x-1)={-(x-1)}^3\Rightarrow (x-1)\{2x-1+{(x-1)}^2\}=0\Rightarrow (x-1){\left(x\right)}^2=0\therefore x_1=0and{x}_2=1$
$\because x_1=0$ is not satisfied the equation (1) then $x_1=0$ is an extraneous root of the equation (1) thus $x_2=1$ is the only root of the original equation.