Question

$\frac{1}{6}( lgx)^{2}- \frac{5}{12}lgx-1=0$

Answer 1

$x\ \textgreater \ 0 \\ fie \\ lgx=t \\ \frac{1}{6}t^2- \frac{5}{12}t -1=0 \\ 2t^2-5t-12=0 \\ t=- \frac{3}{2} ;t=4 \\ revenim \\ lgx=4;x=10^4 \\ lgx=- \frac{3}{2};x= \frac{1}{ \sqrt{10^3} }$

Answer 2

$\displaystyle \mathtt{ \frac{1}{6}(lgx)^2- \frac{5}{12}lgx-1=0 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~C.E.~x\ \textgreater \ 0 }~\\ \\ \mathtt{~~~~~~~~~~lgx=t}\\ \\ \mathtt{ \frac{1}{6}t^2- \frac{5}{12}t-1=0}\\ \\ \mathtt{2t^2-5t-12=0}\\ \\ \mathtt{a=2,~b=-5,~c=-12}\\ \\ \mathtt{\Delta=b^2-4ac=(-5)^2-4 \cdot 2 \cdot (-12)=25+96=121\ \textgreater \ 0}$

$\displaystyle \mathtt{x_1= \frac{-b+ \sqrt{\Delta} }{2a}= \frac{-(-5)+ \sqrt{121} }{2 \cdot 2}= \frac{5+11}{4}= \frac{16}{4}=4}\\ \\ \mathtt{x_2= \frac{-b- \sqrt{\Delta} }{2a}= \frac{-(-5)- \sqrt{121} }{2 \cdot 2}= \frac{5-11}{4}=- \frac{6}{4}=- \frac{3}{2} } \\ \\ \mathtt{lgx=4 \Rightarrow x=10^4 \Rightarrow x=10000 } \\ \\ \mathtt{lgx=- \frac{3}{2} \Rightarrow x=10^{- \frac{3}{2}} \Rightarrow x= \frac{1}{10^{ \frac{3}{2} }} \Rightarrow x= \frac{1}{ \sqrt{10^3} } \Rightarrow x= \frac{ \sqrt{10} }{100} }$

$\displaystyle \mathtt{S=\left\{ \frac{ \sqrt{10} }{100};10000\right\} }$