Страница 71 номер 3.3.11, ГДЗ по алгебре за 7, 8 и 9 класс к задачнику Ткачевой
Решить уравнение: 1) \(x^2 - 42 = x\); 2) \(x^2 - 56 = x\); 3) \(x^2 - x = 30\); 4) \(x^2 - 8x = 9\); 5) \(2x^2 = x + 21\); 6) \(3x^2 = 10 - x\); 7) \(-9x^2 = 2 - 17x\); 8) \(-7x^2 = 6 - 23x\); 9) \(25x^2 + 4 = 20x\); 10) \(16x^2 = 8x - 5\).
1) \(x^2 - 42 = x\), \(x^2 - x - 42 = 0\): \(D = (-1)^2 - 4 \cdot 1 \cdot (-42) = 1 + 168 = 169\), \(D > 0\); \(x = \frac{1 \pm \sqrt{169}}{2} = \frac{1 \pm 13}{2}\), отсюда \(x_1 = -6\), \(x_2 = 7\);
2) \(x^2 - 56 = x\), \(x^2 - x - 56 = 0\): \(D = (-1)^2 - 4 \cdot 1 \cdot (-56) = 1 + 224 = 225\), \(D > 0\); \(x = \frac{1 \pm \sqrt{225}}{2} = \frac{1 \pm 15}{2}\), отсюда \(x_1 = -7\), \(x_2 = 8\);
3) \(x^2 - x = 30\), \(x^2 - x - 30 = 0\): \(D = (-1)^2 - 4 \cdot 1 \cdot (-30) = 1 + 120 = 121\), \(D > 0\); \(x = \frac{1 \pm \sqrt{121}}{2} = \frac{1 \pm 11}{2}\), отсюда \(x_1 = -5\), \(x_2 = 6\);
4) \(x^2 - 8x = 9\), \(x^2 - 8x - 9 = 0\): \(D = (-8)^2 - 4 \cdot 1 \cdot (-9) = 64 + 36 = 100\), \(D > 0\); \(x = \frac{8 \pm \sqrt{100}}{2} = \frac{8 \pm 10}{2}\), отсюда \(x_1 = -1\), \(x_2 = 9\);
5) \(2x^2 = x + 21\), \(2x^2 - x - 21 = 0\): \(D = (-1)^2 - 4 \cdot 2 \cdot (-21) = 1 + 168 = 169\), \(D > 0\); \(x = \frac{1 \pm \sqrt{169}}{4} = \frac{1 \pm 13}{4}\), отсюда \(x_1 = -3\), \(x_2 = 3{,}5\);
6) \(3x^2 = 10 - x\), \(3x^2 + x - 10 = 0\): \(D = 1^2 - 4 \cdot 3 \cdot (-10) = 1 + 120 = 121\), \(D > 0\); \(x = \frac{-1 \pm \sqrt{121}}{6} = \frac{-1 \pm 11}{6}\), отсюда \(x_1 = -2\), \(x_2 = 1\frac{2}{3}\);
7) \(-9x^2 = 2 - 17x\), \(-9x^2 + 17x - 2 = 0\); умножим обе части на \(-1\): \(9x^2 - 17x + 2 = 0\); \(D = (-17)^2 - 4 \cdot 9 \cdot 2 = 289 - 72 = 217\), \(D > 0\); \(x = \frac{17 \pm \sqrt{217}}{18}\), отсюда \(x_1 = \frac{17 - \sqrt{217}}{18}\), \(x_2 = \frac{17 + \sqrt{217}}{18}\);
8) \(-7x^2 = 6 - 23x\), \(-7x^2 + 23x - 6 = 0\); умножим обе части на \(-1\): \(7x^2 - 23x + 6 = 0\); \(D = (-23)^2 - 4 \cdot 7 \cdot 6 = 529 - 168 = 361\), \(D > 0\); \(x = \frac{23 \pm \sqrt{361}}{14} = \frac{23 \pm 19}{14}\), отсюда \(x_1 = \frac{2}{7}\), \(x_2 = 3\);
9) \(25x^2 + 4 = 20x\), \(25x^2 - 20x + 4 = 0\): \(D = (-20)^2 - 4 \cdot 25 \cdot 4 = 400 - 400 = 0\); \(x = \frac{20 \pm 0}{50} = 0{,}4\);
10) \(16x^2 = 8x - 5\), \(16x^2 - 8x + 5 = 0\): \(D = (-8)^2 - 4 \cdot 16 \cdot 5 = 64 - 320 = -256\), \(D < 0\) — корней нет.
Ответ: 1) \(-6\); 7; 2) \(-7\); 8; 3) \(-5\); 6; 4) \(-1\); 9; 5) \(-3\); \(3{,}5\); 6) \(-2\); \(1\frac{2}{3}\); 7) \(\frac{17 - \sqrt{217}}{18}\); \(\frac{17 + \sqrt{217}}{18}\); 8) \(\frac{2}{7}\); 3; 9) \(0{,}4\); 10) корней нет.
Решить уравнение: 1) \(x^2 - 42 = x\); 2) \(x^2 - 56 = x\); 3) \(x^2 - x = 30\); 4) \(x^2 - 8x = 9\); 5) \(2x^2 = x + 21\); 6) \(3x^2 = 10 - x\); 7) \(-9x^2 = 2 - 17x\); 8) \(-7x^2 = 6 - 23x\); 9) \(25x^2 + 4 = 20x\); 10) \(16x^2 = 8x - 5\).
Перенесём все слагаемые в левую часть, приведём уравнение к виду \(ax^2 + bx + c = 0\), вычислим дискриминант \(D = b^2 - 4ac\) и найдём корни по формуле \(x = \frac{-b \pm \sqrt{D}}{2a}\).
1) \(x^2 - 42 = x\), \(x^2 - x - 42 = 0\): \(D = (-1)^2 - 4 \cdot 1 \cdot (-42) = 1 + 168 = 169\), \(D > 0\); \(x = \frac{1 \pm \sqrt{169}}{2} = \frac{1 \pm 13}{2}\), отсюда \(x_1 = -6\), \(x_2 = 7\);
2) \(x^2 - 56 = x\), \(x^2 - x - 56 = 0\): \(D = (-1)^2 - 4 \cdot 1 \cdot (-56) = 1 + 224 = 225\), \(D > 0\); \(x = \frac{1 \pm \sqrt{225}}{2} = \frac{1 \pm 15}{2}\), отсюда \(x_1 = -7\), \(x_2 = 8\);
3) \(x^2 - x = 30\), \(x^2 - x - 30 = 0\): \(D = (-1)^2 - 4 \cdot 1 \cdot (-30) = 1 + 120 = 121\), \(D > 0\); \(x = \frac{1 \pm \sqrt{121}}{2} = \frac{1 \pm 11}{2}\), отсюда \(x_1 = -5\), \(x_2 = 6\);
4) \(x^2 - 8x = 9\), \(x^2 - 8x - 9 = 0\): \(D = (-8)^2 - 4 \cdot 1 \cdot (-9) = 64 + 36 = 100\), \(D > 0\); \(x = \frac{8 \pm \sqrt{100}}{2} = \frac{8 \pm 10}{2}\), отсюда \(x_1 = -1\), \(x_2 = 9\);
5) \(2x^2 = x + 21\), \(2x^2 - x - 21 = 0\): \(D = (-1)^2 - 4 \cdot 2 \cdot (-21) = 1 + 168 = 169\), \(D > 0\); \(x = \frac{1 \pm \sqrt{169}}{4} = \frac{1 \pm 13}{4}\), отсюда \(x_1 = -3\), \(x_2 = 3{,}5\);
6) \(3x^2 = 10 - x\), \(3x^2 + x - 10 = 0\): \(D = 1^2 - 4 \cdot 3 \cdot (-10) = 1 + 120 = 121\), \(D > 0\); \(x = \frac{-1 \pm \sqrt{121}}{6} = \frac{-1 \pm 11}{6}\), отсюда \(x_1 = -2\), \(x_2 = 1\frac{2}{3}\);
7) \(-9x^2 = 2 - 17x\), \(-9x^2 + 17x - 2 = 0\); умножим обе части на \(-1\): \(9x^2 - 17x + 2 = 0\). Тогда \(D = (-17)^2 - 4 \cdot 9 \cdot 2 = 289 - 72 = 217\), \(D > 0\); \(x = \frac{17 \pm \sqrt{217}}{18}\), отсюда \(x_1 = \frac{17 - \sqrt{217}}{18}\), \(x_2 = \frac{17 + \sqrt{217}}{18}\);
8) \(-7x^2 = 6 - 23x\), \(-7x^2 + 23x - 6 = 0\); умножим обе части на \(-1\): \(7x^2 - 23x + 6 = 0\). Тогда \(D = (-23)^2 - 4 \cdot 7 \cdot 6 = 529 - 168 = 361\), \(D > 0\); \(x = \frac{23 \pm \sqrt{361}}{14} = \frac{23 \pm 19}{14}\), отсюда \(x_1 = \frac{2}{7}\), \(x_2 = 3\);
9) \(25x^2 + 4 = 20x\), \(25x^2 - 20x + 4 = 0\): \(D = (-20)^2 - 4 \cdot 25 \cdot 4 = 400 - 400 = 0\); \(x = \frac{20 \pm 0}{50} = 0{,}4\);
10) \(16x^2 = 8x - 5\), \(16x^2 - 8x + 5 = 0\): \(D = (-8)^2 - 4 \cdot 16 \cdot 5 = 64 - 320 = -256\), \(D < 0\) — уравнение не имеет корней.
Ответ: 1) \(-6\); 7; 2) \(-7\); 8; 3) \(-5\); 6; 4) \(-1\); 9; 5) \(-3\); \(3{,}5\); 6) \(-2\); \(1\frac{2}{3}\); 7) \(\frac{17 - \sqrt{217}}{18}\); \(\frac{17 + \sqrt{217}}{18}\); 8) \(\frac{2}{7}\); 3; 9) \(0{,}4\); 10) корней нет.
