ГДЗ по алгебре 8 класс Макарычев Задание 660

\[\boxed{\text{660\ (660).}\text{\ }\text{Еуроки\ -\ ДЗ\ без\ мороки}}\]

\[Пусть\ x_{1},\ x_{2},\ x_{3},\ x_{4},\ x_{5} - пять\ \]

\[последовательных\ целых\ \]

\[чисел.\]

\[x_{1}^{2} + x_{2}^{2} + x_{3}^{2} = x_{4}^{2} + x_{5}^{2}\]

\[x_{2} = x_{1} + 1,\]

\[\text{\ \ }x_{3} = x_{1} + 2,\]

\[\text{\ \ }x_{4} = x_{1} + 3,\ \ x_{5} = x_{1} + 4.\]

\[Составим\ уравнение:\]

\[x_{1}^{2} + \left( x_{1} + 1 \right)^{2} + \left( x_{1} + 2 \right)^{2} =\]

\[= \left( x_{1} + 3 \right)^{2} + \left( x_{1} + 4 \right)^{2}\]

\[x_{1}^{2} + x_{1}^{2} + 2x_{1} + 1 + x_{1}^{2} + 4x_{1} + 4 =\]

\[= x_{1}^{2} + 6x + 9 + x_{1}^{2} + 8x_{1} + 16\]

\[3x_{1}^{2} + 6x_{1} + 5 =\]

\[= 2x_{1}^{2} + 14x_{1} + 25\]

\[x_{1}^{2} - 8x_{1} - 20 = 0\]

\[D = 64 + 80 = 144\]

\[x_{1,2} = \frac{8 \pm 12}{2} = 10;\ - 2\]

\[то\ есть:\]

\[x_{1} = 10,\ x_{2} = 11,\ x_{3} = 12,\ \]

\[x_{4} = 13,\ x_{5} = 14\ или\]

\[x_{1} = - 2,\ x_{2} = - 1,\ x_{3} = 0,\ \]

\[x_{4} = 1,\ x_{5} = 2.\ \]

\[Ответ:10,\ 11,\ 12,\ 13,\ 14\ \ или\]

\[\text{\ \ }( - 2),\ - 1,\ 0,\ 1,\ 2.\]