\(\dfrac{1}{{{{\left( {x-2} \right)}^2}}}-\dfrac{1}{{x-2}}-6 = 0.\)
Пусть \(\dfrac{1}{{x-2}} = t\). Тогда уравнение примет вид:
\({t^2}-t-6 = 0\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{t = 3,\,\,\,}\\{t = -2.}\end{array}} \right.\)
Вернёмся к прежней переменной:
\(\left[ {\begin{array}{*{20}{c}}{\dfrac{1}{{x-2}} = 3,\,\,}\\{\dfrac{1}{{x-2}} = -2}\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{\left[ {\begin{array}{*{20}{c}}{x-2 = \dfrac{1}{3},\,\,\,}\\{x-2 = — \dfrac{1}{2},}\end{array}} \right.}\\{x \ne 2\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\end{array}} \right.\,\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{\left[ {\begin{array}{*{20}{c}}{x = \dfrac{7}{3},}\\{x = \dfrac{3}{2},}\end{array}} \right.}\\{x \ne 2\,\,\,}\end{array}} \right.\,\,\,\,\,\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{x = \dfrac{7}{3},\,}\\{x = \dfrac{3}{2}.}\end{array}} \right.\)
Ответ: \(\dfrac{3}{2};\;\;\dfrac{7}{3}.\)