Страница 32 номер 2.2.28, ГДЗ по алгебре за 7, 8 и 9 класс к задачнику Ткачевой
Упростить выражение: 1) \((a + b)^2 - (a - b)^2\); 2) \((x - y)^2 - (x + y)^2\); 3) \((2a + 3b)^2 + (2a - 3b)^2\); 4) \((3n - 4m)^2 + (3n + 4m)^2\); 5) \((3x - 1)^2 - (3x - 1)(3x + 1)\); 6) \((2c + 1)^2 - (2c - 1)(2c + 1)\); 7) \((ab + 2)(2 - ab) - (ab + 2)^2\); 8) \((xy + 3)(3 - xy) - (3 - xy)^2\).
1) \[\begin{aligned} &(a + b)^2 - (a - b)^2 = (a^2 + 2ab + b^2) - (a^2 - 2ab + b^2) = {} \\ &= a^2 + 2ab + b^2 - a^2 + 2ab - b^2 = 4ab; \end{aligned}\]
2) \[\begin{aligned} &(x - y)^2 - (x + y)^2 = (x^2 - 2xy + y^2) - (x^2 + 2xy + y^2) = {} \\ &= x^2 - 2xy + y^2 - x^2 - 2xy - y^2 = -4xy; \end{aligned}\]
3) \[\begin{aligned} &(2a + 3b)^2 + (2a - 3b)^2 = (4a^2 + 12ab + 9b^2) + (4a^2 - 12ab + 9b^2) = {} \\ &= 4a^2 + 12ab + 9b^2 + 4a^2 - 12ab + 9b^2 = 8a^2 + 18b^2; \end{aligned}\]
4) \[\begin{aligned} &(3n - 4m)^2 + (3n + 4m)^2 = (9n^2 - 24mn + 16m^2) + (9n^2 + 24mn + 16m^2) = {} \\ &= 9n^2 - 24mn + 16m^2 + 9n^2 + 24mn + 16m^2 = 18n^2 + 32m^2; \end{aligned}\]
5) \[\begin{aligned} &(3x - 1)^2 - (3x - 1)(3x + 1) = (9x^2 - 6x + 1) - (9x^2 - 1) = {} \\ &= 9x^2 - 6x + 1 - 9x^2 + 1 = -6x + 2; \end{aligned}\]
6) \[\begin{aligned} &(2c + 1)^2 - (2c - 1)(2c + 1) = (4c^2 + 4c + 1) - (4c^2 - 1) = {} \\ &= 4c^2 + 4c + 1 - 4c^2 + 1 = 4c + 2; \end{aligned}\]
7) \[\begin{aligned} &(ab + 2)(2 - ab) - (ab + 2)^2 = (2 + ab)(2 - ab) - (ab + 2)^2 = {} \\ &= (4 - a^2b^2) - (a^2b^2 + 4ab + 4) = {} \\ &= 4 - a^2b^2 - a^2b^2 - 4ab - 4 = -2a^2b^2 - 4ab; \end{aligned}\]
8) \[\begin{aligned} &(xy + 3)(3 - xy) - (3 - xy)^2 = (3 + xy)(3 - xy) - (3 - xy)^2 = {} \\ &= (9 - x^2y^2) - (9 - 6xy + x^2y^2) = {} \\ &= 9 - x^2y^2 - 9 + 6xy - x^2y^2 = 6xy - 2x^2y^2. \end{aligned}\]
Ответ: 1) \(4ab\); 2) \(-4xy\); 3) \(8a^2 + 18b^2\); 4) \(18n^2 + 32m^2\); 5) \(-6x + 2\); 6) \(4c + 2\); 7) \(-2a^2b^2 - 4ab\); 8) \(6xy - 2x^2y^2\).
Упростить выражение: 1) \((a + b)^2 - (a - b)^2\); 2) \((x - y)^2 - (x + y)^2\); 3) \((2a + 3b)^2 + (2a - 3b)^2\); 4) \((3n - 4m)^2 + (3n + 4m)^2\); 5) \((3x - 1)^2 - (3x - 1)(3x + 1)\); 6) \((2c + 1)^2 - (2c - 1)(2c + 1)\); 7) \((ab + 2)(2 - ab) - (ab + 2)^2\); 8) \((xy + 3)(3 - xy) - (3 - xy)^2\).
Применяются формулы сокращённого умножения \((a + b)^2 = a^2 + 2ab + b^2\), \((a - b)^2 = a^2 - 2ab + b^2\) и \((a - b)(a + b) = a^2 - b^2\). Если перед скобками стоит знак «минус», то знаки всех членов, стоящих в скобках, меняются на противоположные.
1) \[\begin{aligned} &(a + b)^2 - (a - b)^2 = (a^2 + 2ab + b^2) - (a^2 - 2ab + b^2) = {} \\ &= a^2 + 2ab + b^2 - a^2 + 2ab - b^2 = 4ab; \end{aligned}\]
2) \[\begin{aligned} &(x - y)^2 - (x + y)^2 = (x^2 - 2xy + y^2) - (x^2 + 2xy + y^2) = {} \\ &= x^2 - 2xy + y^2 - x^2 - 2xy - y^2 = -4xy; \end{aligned}\]
3) \[\begin{aligned} &(2a + 3b)^2 + (2a - 3b)^2 = (4a^2 + 12ab + 9b^2) + (4a^2 - 12ab + 9b^2) = {} \\ &= 4a^2 + 12ab + 9b^2 + 4a^2 - 12ab + 9b^2 = 8a^2 + 18b^2; \end{aligned}\]
4) \[\begin{aligned} &(3n - 4m)^2 + (3n + 4m)^2 = (9n^2 - 24mn + 16m^2) + (9n^2 + 24mn + 16m^2) = {} \\ &= 9n^2 - 24mn + 16m^2 + 9n^2 + 24mn + 16m^2 = 18n^2 + 32m^2; \end{aligned}\]
5) \[\begin{aligned} &(3x - 1)^2 - (3x - 1)(3x + 1) = (9x^2 - 6x + 1) - (9x^2 - 1) = {} \\ &= 9x^2 - 6x + 1 - 9x^2 + 1 = -6x + 2; \end{aligned}\]
6) \[\begin{aligned} &(2c + 1)^2 - (2c - 1)(2c + 1) = (4c^2 + 4c + 1) - (4c^2 - 1) = {} \\ &= 4c^2 + 4c + 1 - 4c^2 + 1 = 4c + 2; \end{aligned}\]
7) \[\begin{aligned} &(ab + 2)(2 - ab) - (ab + 2)^2 = (2 + ab)(2 - ab) - (ab + 2)^2 = {} \\ &= (4 - a^2b^2) - (a^2b^2 + 4ab + 4) = {} \\ &= 4 - a^2b^2 - a^2b^2 - 4ab - 4 = -2a^2b^2 - 4ab; \end{aligned}\]
8) \[\begin{aligned} &(xy + 3)(3 - xy) - (3 - xy)^2 = (3 + xy)(3 - xy) - (3 - xy)^2 = {} \\ &= (9 - x^2y^2) - (9 - 6xy + x^2y^2) = {} \\ &= 9 - x^2y^2 - 9 + 6xy - x^2y^2 = 6xy - 2x^2y^2. \end{aligned}\]
Ответ: 1) \(4ab\); 2) \(-4xy\); 3) \(8a^2 + 18b^2\); 4) \(18n^2 + 32m^2\); 5) \(-6x + 2\); 6) \(4c + 2\); 7) \(-2a^2b^2 - 4ab\); 8) \(6xy - 2x^2y^2\).
