Question

Решите неравенство (log2^2 x - 2log2 x )^2 + 36log2 x + 45< 18log2^2 x

Answer 1

$\displaystyle (log_2^2x-2log_2x)^2+36log_2x+45\ \textless \ 18log_2^2x$

$\displaystyle (log_2^2x-2log_2x)^2-18(log_2^2x-2log_2x)+45\ \textless \ 0$

$\displaystyle Log_2^2x-2log_2x=t$

$\displaystyle t^2-18t+45\ \textless \ 0$

$\displaystyle D=324-180=144=12^2 t_P1=15; t_2=3$

__+_____-_____+____ 
         3                15

$\displaystyle \left \{ {{Log_2^2x-2log_2x\ \textless \ 15} \atop {log_2^2x-2log^2x\ \textgreater \ 3}} \right.$

1) 
$\displaystyle log_2^2x-2log_2x-15\ \textless \ 0$

$\displaystyle D=4+64=64=8^2$

_+____-_____+____
    -3              5

$\displaystyle -3\ \textless \ log_2x\ \textless \ 5$

$\displaystyle \frac{1}{2^3}\ \textless \ x\ \textless \ 2^5$

2)
$\displaystyle log_2^2x-2log_2x-3\ \textgreater \ 0$

$\displaystyle D=4^2$

 __+___-____+___
       -1        3

$\displaystyle \left \{ {{log_2x\ \textgreater \ 3} \atop {log_2x\ \textless \ -1}} \right.$

$\displaystyle \left \{ {{x\ \textless \ \frac{1}{2}} \atop {x\ \textgreater \ 2^5}} \right.$

Объединим все промежутки с учетом ОДЗ х>0

   \\\\\\\\\\\\\\\\\\\\\             \\\\\\\\\\\\\\\\\\\\\\\\\\\\
            //////////////////////////////////////
  0      1/8      1/2          8          32

ОТВЕТ: (1/8; 1/2) ∪(8;32)