Нужно воспользоваться формулой разности квадратов практически во всех примерах: (a - b)(a + b) = a² - b².
Выполните умножение:
1) 5b(b - 1)(b + 1) = 5b(b² - 1) = 5b³ - 5b;
2) (c + 2)(c - 2) · 8c² = (c² - 4) · 8c² = 8c⁴ - 32c²;
3) (m - 10)(m² + 100)(m + 10) = (m - 10)(m + 10)(m² + 100) =
= (m² - 100)(m² + 100) = m⁴ - 10 000;
4) (a² + 1)(a² - 1)(a⁴ + 1) = (a⁴ - 1)(a⁴ + 1) = a⁸ - 1;
Упростите выражение:
1) (x + 1)(x - 1) - (x + 5)(x - 5) + (x + 1)(x - 5) = x² - 1 - (x² - 25) + x² - 5x + x - 5 = x² - 1 - x² + 25 + x² - 4x - 5 = x² - 4x + 19;
2) 81a⁸ - (3a² - b³)(9a⁴ + b⁶)(3a² + b³) = 81a⁸ - (3a² - b³)(3a² + b³)(9a⁴ + b⁶) = 81a⁸ - (9a⁴ - b⁶)(9a⁴ + b⁶) = 81a⁸ - (81a⁸ - b¹²) = 81a⁸ - 81a⁸ + b¹² = b¹².
a) (2x - y)(x + 2y) = -3
1) 2x - y = 3 => y = 2x - 3
x + 2y = -1 => x + 2(2x - 3) = - 1 =>
x + 4x - 6 = -1 => 5x = 5 => x = 1
y = 2x - 3 = 2 - 3 = -1.
Первое решение (x, y) = (1, -1)
2) 2x - y = - 3 => y = 2x + 3
x + 2y = 1 => x +2(2x + 3) = 1 =>
x + 4x + 6 = 1 => 5x = - 5 => x = -1
y = 2x + 3 = -2 + 3 = 1.
Второе решение (x, y) = (-1, 1)
3) 2x - y = 1 => y = 2x - 1
x + 2y = -3 => x + 2(2x - 1) = -3 =>
x + 4x - 2 = -3 => 5x = -1 => x = - 1/5 - нецелое.
4) 2x - y = -1 => y = 2x + 1
x + 2y = 3 => x + 2(2x + 1) = 3 =>
x + 4x + 2 = 3 => 5x = 1 => x = 1/5 - нецелое.
Всего два решения (x, y) = (1, -1) и (x, y) = (-1, 1)
б) 2x² + xy - y² = -3
x² - y² + x² + xy = -3
(x - y)(x + y) + x(x + y) = -3
(x + y)(x - y + x) = -3
(x + y)(2x - y) = -3
1) x + y = 3 => y = 3 - x
2x - y = -1 => 2x - 3 + x = -1 =>
3x - 3 = -1 => 3x = 2 => x = 2/3 - нецелое
2) x + y = 1 => y = 1 - x
2x - y = -3 => 2x - 1 + x = -3 =>
3x -1 = -3 => 3x = -2 => x = -2/3 - нецелое
3) x + y = -3 => y = -3 - x
2x - y = 1 => 2x +3 + x = 1 =>
3x + 3 = 1 = 3x = -2 => x = -2/3 - нецелое
4) x + y = -1 = y = -1 - x
2x - y = 3 => 2x +1 + x = 3 =>
3x + 1 = 3 => 3x = 2 => x = 2/3 - нецелое.
Второе уравнение не имеет решений в целых числах.