13А. Найдите все значения параметра a, при каждом из которых корни уравнения \({x^2}-2a\,x + {a^2}-a = 0\) принадлежат отрезку \(\left[ {-2;\,6} \right]\).
\(\left\{ {\begin{array}{*{20}{c}}{D \ge 0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{-2 \le {x_B} = -\frac{b}{{2a}} \le 6,}\\{f\left( {-2} \right) \ge 0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{f\left( 6 \right) \ge 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\end{array}} \right.\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{{{\left( {-2a} \right)}^2}-4\left( {{a^2}-a} \right) \ge 0,}\\{-2 \le a \le 6,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{4 + 4a + {a^2}-a \ge 0,\,\,\,\,\,\,\,\,\,}\\{36-12a + {a^2}-a \ge 0\,\,\,\,\,\,}\end{array}} \right.\,\,\,\,\,\,\,\, \Leftrightarrow \) \( \Leftrightarrow \,\,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{4a \ge 0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{-2 \le a \le 6,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{{a^2} + 3a + 4 \ge 0,\,\,\,}\\{{a^2}-13a + 36 \ge 0}\end{array}} \right.\,\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{a \in \left[ {0;\,\infty } \right),\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{a \in \left[ {-2;\,6} \right],\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{a \in R,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{a \in \,\left( {-\infty ;\,4} \right] \cup \left[ {9;\infty } \right)}\end{array}} \right.\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,a\, \in \,\left[ {0;4} \right].\) Таким образом, корни уравнения будут принадлежать отрезку \(\left[ {-2;\,6} \right]\) при \(a \in \left[ {0;\,4} \right].\) ОТВЕТ: \(\left[ {0;\,4} \right].\)
Введём функцию \(f\left( x \right) = {x^2}-2ax + {a^2}-a\) графиком которой является парабола ветвями вверх. Для того чтобы корни уравнения принадлежали отрезку\(\left[ {-2;\,6} \right]\), необходимо выполнение следующих условий (см. рис.):