Алгебра
Английский
Биология
География
Геометрия
История
Русский
Обществознание
Физика
Химия
Информатика
Литература
МатематикаРешебник по алгебре 8 класс Мерзляк ФГОС Задание 183
Задание 183
\[\boxed{\text{183\ (183).}\text{\ }\text{Еуроки\ -\ ДЗ\ без\ мороки}}\]
\[1)\left( \frac{b}{a^{2} - ab} - \frac{2}{a - b} - \frac{a}{b^{2} - ab} \right)\ :\]
\[:\frac{a^{2} - b^{2}}{4ab} = \frac{4}{a + b}\]
\[Упростим\ левую\ часть\ \]
\[равенства:\]
\[\left( \frac{b^{\backslash b}}{a(a - b)} - \frac{2^{\backslash ab}}{a - b} + \frac{a^{\backslash a}}{b(a - b)} \right)\ :\]
\[:\frac{a^{2} - b^{2}}{4ab} = \frac{4}{a + b}\]
\[\frac{b^{2} - 2ab + a^{2}}{\text{ab}(a - b)}\ :\frac{a^{2} - b^{2}}{4ab} = \frac{4}{a + b}\]
\[\frac{(b - a)^{2} \cdot 4ab}{ab(a - b)(a - b)(a + b)} = \frac{4}{a + b}\]
\[\frac{4}{a + b} = \frac{4}{a + b}.\]
\[Тождество\ доказано.\]
\[2)\ \frac{(a - b)^{2}}{a} \cdot\]
\[\cdot \left( \frac{a^{\backslash a + b}}{(a - b)^{2}} + \frac{a^{\backslash b - a}}{b^{2} - a^{2}} \right) +\]
\[+ \frac{3a + b}{a + b} = 3\]
\[Упростим\ левую\ часть\ \]
\[равенства:\]
\[\frac{(a - b)^{2}}{a} \cdot \left( \frac{a^{2} + ab - a^{2} + ab}{(a - b)^{2}(a + b)} \right) +\]
\[+ \frac{3a + b}{a + b} = 3\]
\[\frac{(a - b)^{2} \cdot 2ab}{a(a - b)^{2}(a + b)} + \frac{3a + b}{a + b} = 3\]
\[\frac{2b}{a + b} + \frac{3a + b}{a + b} = 3\]
\[\frac{2b + 3a + b}{a + b} = 3\]
\[\frac{3(a + b)}{(a + b)} = 3\]
\[3 = 3.\]
\[Тождество\ доказано.\]
\[\boxed{\text{183.}\text{\ }\text{Еуроки\ -\ ДЗ\ без\ мороки}}\]
\[1)\frac{3x^{2} - 27}{4x^{2} + 2} \cdot\]
\[\cdot \left( \frac{6x + 1^{\backslash x + 3}}{x - 3} + \frac{6x - 1^{\backslash x - 3}}{x + 3} \right) =\]
\[= \frac{3x^{2} - 27}{4x^{2} + 2} \cdot\]
\[\cdot \left( \frac{6x^{2} + x + 18x + 3 + 6x^{2} - x - 18x + 3}{(x - 3)(x + 3)} \right) =\]
\[= \frac{3 \cdot \left( x^{2} - 9 \right)}{2 \cdot \left( 2x^{2} + 1 \right)} \cdot \frac{12x^{2} + 6}{(x - 3)(x + 3)} =\]
\[= \frac{3(x - 3)(x + 3) \cdot 6 \cdot \left( 2x^{2} + 1 \right)}{2\left( 2x^{2} + 1 \right)(x - 3)(x + 3)} =\]
\[= \frac{18}{2} = 9\]
\[2)\ \frac{3}{2a - 3} - \frac{8a^{3} - 18a}{4a^{2} + 9} \cdot\]
\[\cdot \left( \frac{2a}{4a^{2} - 12a + 9} - \frac{3}{4a^{2} - 9} \right) =\]
\[= \frac{3}{2a - 3} - \frac{8a^{3} - 18a}{4a^{2} + 9} \cdot\]
\[\cdot \left( \frac{2a^{\backslash 2a + 3}}{(2a - 3)^{2}} - \frac{3^{\backslash 2a - 3}}{(2a - 3)(2a + 3)} \right) =\]
\[= \frac{3}{2a - 3} - \frac{8a^{3} - 18a}{4a^{2} + 9} \cdot\]
\[\cdot \frac{4a^{2} + 6a - 6a + 9}{(2a - 3) \cdot (2a + 3)} =\]
\[= \frac{3}{2a - 3} - \frac{2a \cdot \left( 4a^{2} - 9 \right)}{4a^{2} + 9} \cdot\]
\[\cdot \frac{4a^{2} + 9}{(2a - 3)^{2}(2a + 3)} =\]
\[= \frac{3}{2a - 3} -\]
\[- \frac{2a(2a - 3)(2a + 3)\left( 4a^{2} + 9 \right)}{\left( 4a^{2} + 9 \right)(2a - 3)^{2}(2a + 3)} =\]
\[= \frac{3}{2a - 3} - \frac{2a}{2a - 3} = \frac{3 - 2a}{2a - 3} =\]
\[= - 1\]
