Question

$2 lg^{3}x+3 lg^{2}x-2lgx=0$

Answer 1

$x\ \textgreater \ 0 \\ lgx(2 lg^{2}x+ 3lgx-2)=0 \\ lgx=0sau2lg^2x+3lgx-2=0 \\ x=1;sau;notam \\ lgx=t \\ 2t^2+3t-2=0 \\ t=-2;t= \frac{1}{2} \\ revenim \\ lgx=-2;x= \frac{1}{100} \\ lgx= \frac{1}{2};x= \sqrt{10}$
S={1;1/100;√10}

Answer 2

$\displaystyle \mathtt{2lg^3x+3lg^2x-2lgx=0 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~C.E.~x\ \textgreater \ 0}\\ \\ \mathtt{~~~~~~~~~~~~lgx=t}\\ \\ \mathtt{2t^3+3t^2-2t=0}\\ \\ \mathtt{t\left(2t^2+3t-2\right)=0}\\ \\ \mathtt{t(t+2)(2t-1)=0}\\ \\ \mathtt{t=0;~~~~~~~~~t+2=0 \Rightarrow t=-2;~~~~~~~~~2t-1=0 \Rightarrow t= \frac{1}{2} }\\ \\ \mathtt{lgx=0 \Rightarrow lgx=lg1\Rightarrow x=1}\\ \\ \mathtt{lgx=-2\Rightarrow lgx=lg\frac{1}{100}\Rightarrow x= \frac{1}{100}}\\ \\ \mathtt{lgx=\frac{1}{2}\Rightarrow lgx=lg \sqrt{10}\Rightarrow x=\sqrt{10}}$
$\displaystyle \mathtt{S=\left\{ \frac{1}{100};1; \sqrt{10}\right\} }$