ГДЗ по алгебре 8 класс Макарычев ФГОС Задание 1004

\[\boxed{\text{1004.\ }\text{еуроки}\text{-}\text{ответы}\text{\ }\text{на}\text{\ }\text{пятёрку}}\]

\[\textbf{а)}\ a² + b² + 4 \geq 2 \cdot (a + b + 1)\]

\[a^{2} + b^{2} + 4 -\]

\[- 2 \cdot (a + b + 1) \geq 0\]

\[a² + b² + 4 - 2a - 2b - 2 \geq 0\]

\[a^{2} - 2a - 2 + b^{2} - 2b + 4 \geq 0\]

\[\left( a^{2} - 2a + 1 \right) - 3 +\]

\[+ \left( b^{2} - 2b + 1 \right) + 3 \geq 0\]

\[(a - 1)^{2} + (b - 1)^{2} \geq 0\]

\[(a - 1)^{2} \geq 0,\ \ (b - 1)^{2} \geq\]

\[\geq 0 \Longrightarrow верно\ при\ любых\ \text{a\ }и\ b.\]

\[\Longrightarrow a^{2} + b^{2} + 4 \geq 2 \cdot\]

\[\cdot (a + b + 1) \Longrightarrow ч.т.д.\]

\[\textbf{б)}\ 4a² + b² > 4 \cdot (a + b - 2)\]

\[4a^{2} + b^{2} - 4 \cdot (a + b - 2) > 0\]

\[4a^{2} + b^{2} - 4a - 4b + 8 > 0\]

\[\left( 4a^{2} - 4a + 1 \right) +\]

\[+ \left( b^{2} - 4b + 4 \right) + 3 \geq 0\]

\[(2a - 1)^{2} + (b - 2)^{2} + 3 \geq 0\]

\[(2a + 1)^{2} \geq 0,\ \ \]

\[(b - 2)^{2} \geq 0,\ \ 3 \geq 0\]

\[верно\ при\ любых\ a\ и\ b.\]

\[\Longrightarrow 4a^{2} + b^{2} > 4 \cdot\]

\[\cdot (a + b - 2) \Longrightarrow ч.т.д.\]