Решебник по алгебре 9 класс Мерзляк Задание 756

 

\[\boxed{\mathbf{756\ (756).\ }Еуроки\ - \ ДЗ\ без\ мороки}\]

\[Запишем:\ \]

\[\ \frac{1}{a + c} = \frac{\frac{1}{b + c} + \frac{1}{a + b}}{2}\text{\ \ }\]

\[\frac{2}{a + c} = \frac{1}{b + c} + \frac{1}{a + b}\]

\[\frac{2}{a + c} = \frac{a + b + b + c}{(b + c)(a + b)}\text{\ \ }\]

\[\frac{2}{a + c} = \frac{a + c + 2b}{(b + c)(a + b)}\text{\ \ }\]

\[\frac{2}{a + c} = \frac{a + c + 2b}{ab + b^{2} + ac + bc}\ \]

\[2 \cdot \left( ab + b^{2} + ac + bc \right) =\]

\[= (a + c)(a + c + 2b)\]

\[2ab + 2b^{2} + 2ac + 2bc =\]

\[= a^{2} + ac + 2ab + ac +\]

\[+ c^{2} + 2bc\]

\[2b^{2} = a^{2} + c^{2}\ или\ \ \ \ \]

\[b^{2} = \frac{a^{2} + c^{2}}{2}\text{\ \ }\]

\[Тогда:\ \ \]

\[a^{2},\ b^{2},\ c^{2} - тоже\ члены\]

\[\ арифметической\ прогрессии.\]

\[\boxed{\mathbf{756.\ }Еуроки\ - \ ДЗ\ без\ мороки}\]

\[1)\ 6ab² - 12ab^{3} =\]

\[= 6ab\left( b - 2b^{2} \right) = 6ab²(1 - 2b)\]

\[2)\ 2a³ - 8a = 2a\left( a^{2} - 4 \right) =\]

\[= 2a(a - 2)(a + 2)\]

\[3)\ 3a²c - 3c^{3} = 3c\left( a^{2} - c^{2} \right) =\]

\[= 3c(a - c)(a + c)\]

\[4)\ 18mn² + 27m²n =\]

\[= 9mn(2n + 3m)\]

\[5)\ 100x² - 1 =\]

\[= (10x - 1)(10x + 1)\]

\[6)\ 2y² - 12y + 18 =\]

\[= 2 \cdot \left( y^{2} - 6y + 9 \right) = 2 \cdot (y - 3)²\]