Задача 3. Решите уравнение \(\frac{1}{{{{\left( {x-2} \right)}^2}}}-\frac{1}{{x-2}}-6 = 0\)
ОТВЕТ: 3/2; 7/3.
\(\frac{1}{{{{\left( {x-2} \right)}^2}}}-\frac{1}{{x-2}}-6 = 0.\) Пусть \(\frac{1}{{x-2}} = t\). 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: \(\left[ {\begin{array}{*{20}{c}}{\frac{1}{{x-2}} = 3,\,\,}\\{\frac{1}{{x-2}} = -2}\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{\left[ {\begin{array}{*{20}{c}}{x-2 = \frac{1}{3},\,\,\,}\\{x-2 = — \frac{1}{2},}\end{array}} \right.}\\{x \ne 2\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\end{array}} \right.\,\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{\left[ {\begin{array}{*{20}{c}}{x = \frac{7}{3},}\\{x = \frac{3}{2},}\end{array}} \right.}\\{x \ne 2\,\,\,}\end{array}} \right.\,\,\,\,\,\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{x = \frac{7}{3},\,}\\{x = \frac{3}{2}.}\end{array}} \right.\) Ответ: \(\frac{3}{2};\;\;\frac{7}{3}.\)