Страница 74 номер 3.3.36, ГДЗ по алгебре за 7, 8 и 9 класс к задачнику Ткачевой
Решить уравнение (3.3.32—3.3.38): 1) \(6x^2 + 5x = 6 - 14x - x^2\); 2) \(7x^2 + 3x = 10 - 8x - x^2\); 3) \(5x^2 - 7 = 15x - 13x^2\); 4) \(13x^2 + 5x = 3x^2 + 2x + 18\).
1) \(6x^2 + 5x = 6 - 14x - x^2\); \[6x^2 + 5x - 6 + 14x + x^2 = 0, \\ 7x^2 + 19x - 6 = 0.\] \(D = 19^2 - 4 \cdot 7 \cdot (-6) = 361 + 168 = 529\), \(x = \frac{-19 \pm 23}{14}\), \(x_1 = -\frac{42}{14} = -3\), \(x_2 = \frac{4}{14} = \frac{2}{7}\);
2) \(7x^2 + 3x = 10 - 8x - x^2\); \[7x^2 + 3x - 10 + 8x + x^2 = 0, \\ 8x^2 + 11x - 10 = 0.\] \(D = 11^2 - 4 \cdot 8 \cdot (-10) = 121 + 320 = 441\), \(x = \frac{-11 \pm 21}{16}\), \(x_1 = -\frac{32}{16} = -2\), \(x_2 = \frac{10}{16} = \frac{5}{8}\);
3) \(5x^2 - 7 = 15x - 13x^2\); \[5x^2 - 7 - 15x + 13x^2 = 0, \\ 18x^2 - 15x - 7 = 0.\] \(D = (-15)^2 - 4 \cdot 18 \cdot (-7) = 225 + 504 = 729\), \(x = \frac{15 \pm 27}{36}\), \(x_1 = -\frac{12}{36} = -\frac{1}{3}\), \(x_2 = \frac{42}{36} = 1\frac{1}{6}\);
4) \(13x^2 + 5x = 3x^2 + 2x + 18\); \[13x^2 + 5x - 3x^2 - 2x - 18 = 0, \\ 10x^2 + 3x - 18 = 0.\] \(D = 3^2 - 4 \cdot 10 \cdot (-18) = 9 + 720 = 729\), \(x = \frac{-3 \pm 27}{20}\), \(x_1 = -\frac{30}{20} = -1{,}5\), \(x_2 = \frac{24}{20} = 1{,}2\).
Ответ: 1) \(-3\); \(\frac{2}{7}\); 2) \(-2\); \(\frac{5}{8}\); 3) \(-\frac{1}{3}\); \(1\frac{1}{6}\); 4) \(-1{,}5\); \(1{,}2\).
Решить уравнение (3.3.32—3.3.38): 1) \(6x^2 + 5x = 6 - 14x - x^2\); 2) \(7x^2 + 3x = 10 - 8x - x^2\); 3) \(5x^2 - 7 = 15x - 13x^2\); 4) \(13x^2 + 5x = 3x^2 + 2x + 18\).
В каждом уравнении перенесём все члены в левую часть, приведём подобные слагаемые и решим полученное квадратное уравнение по формуле корней.
1) \(6x^2 + 5x = 6 - 14x - x^2\); \[6x^2 + 5x - 6 + 14x + x^2 = 0, \\ 7x^2 + 19x - 6 = 0.\] \(D = 19^2 - 4 \cdot 7 \cdot (-6) = 361 + 168 = 529\), \(x = \frac{-19 \pm 23}{14}\), \(x_1 = -\frac{42}{14} = -3\), \(x_2 = \frac{4}{14} = \frac{2}{7}\);
2) \(7x^2 + 3x = 10 - 8x - x^2\); \[7x^2 + 3x - 10 + 8x + x^2 = 0, \\ 8x^2 + 11x - 10 = 0.\] \(D = 11^2 - 4 \cdot 8 \cdot (-10) = 121 + 320 = 441\), \(x = \frac{-11 \pm 21}{16}\), \(x_1 = -\frac{32}{16} = -2\), \(x_2 = \frac{10}{16} = \frac{5}{8}\);
3) \(5x^2 - 7 = 15x - 13x^2\); \[5x^2 - 7 - 15x + 13x^2 = 0, \\ 18x^2 - 15x - 7 = 0.\] \(D = (-15)^2 - 4 \cdot 18 \cdot (-7) = 225 + 504 = 729\), \(x = \frac{15 \pm 27}{36}\), \(x_1 = -\frac{12}{36} = -\frac{1}{3}\), \(x_2 = \frac{42}{36} = 1\frac{1}{6}\);
4) \(13x^2 + 5x = 3x^2 + 2x + 18\); \[13x^2 + 5x - 3x^2 - 2x - 18 = 0, \\ 10x^2 + 3x - 18 = 0.\] \(D = 3^2 - 4 \cdot 10 \cdot (-18) = 9 + 720 = 729\), \(x = \frac{-3 \pm 27}{20}\), \(x_1 = -\frac{30}{20} = -1{,}5\), \(x_2 = \frac{24}{20} = 1{,}2\).
Ответ: 1) \(-3\); \(\frac{2}{7}\); 2) \(-2\); \(\frac{5}{8}\); 3) \(-\frac{1}{3}\); \(1\frac{1}{6}\); 4) \(-1{,}5\); \(1{,}2\).
