Страница 204 номер 579, ГДЗ по алгебре за 9 класс к учебнику Алимова с ответами и решением
Упростить выражение (578—582). 1) \(\frac{a - 3}{a + 3} - \frac{a^2 + 27}{a^2 - 9}\); 2) \(\frac{a^2 + 12}{a^2 - 4} - \frac{a + 3}{a - 2}\); 3) \(\frac{a + 1}{a^2 - ax} - \frac{x + 1}{a^2 - x^2}\); 4) \(\frac{3 - a}{ab - a^2} - \frac{3 - b}{b^2 - a^2}\).
1) \(\frac{a - 3}{a + 3} - \frac{a^{2} + 27}{a^{2} - 9} = \frac{(a - 3)^{2} - a^{2} - 27}{(a - 3)(a + 3)} = \frac{a^{2} - 6a + 9 - a^{2} - 27}{(a - 3)(a + 3)} = \frac{-6(a + 3)}{(a - 3)(a + 3)} = \frac{6}{3 - a}\);
2) \(\frac{a^{2} + 12}{a^{2} - 4} - \frac{a + 3}{a - 2} = \frac{a^{2} + 12 - (a + 3)(a + 2)}{(a - 2)(a + 2)} = \frac{12 - 2a - 3a - 6}{(a - 2)(a + 2)} = \frac{6 - 5a}{a^{2} - 4}\);
3) \(\frac{a + 1}{a^{2} - ax} - \frac{x + 1}{a^{2} - x^{2}} = \frac{(a + 1)(a + x) - (x + 1)a}{a(a - x)(a + x)} = \frac{a^{2} + ax + a + x - ax - a}{a(a - x)(a + x)} = \frac{a^{2} + x}{a(a^{2} - x^{2})}\);
4) \(\frac{3 - a}{ab - a^{2}} - \frac{3 - b}{b^{2} - a^{2}} = \frac{(3 - a)(b + a) - (3 - b)a}{a(b - a)(b + a)} = \frac{3b + 3a - ab - a^{2} - 3a + ab}{a(b - a)(b + a)} = \frac{3b - a^{2}}{a\left(b^{2} - a^{2}\right)}\).
