Упр.499 ГДЗ Колягин Ткачёва 7 класс (Алгебра)
Разложить на множители:
- $$7(y-3)-a(3-y)$$
- $$6(a-2)+a(2-a)$$
- $$b^2(a-1)-c(1-a)$$
- $$a^2(m-2)+b(2-m)$$
Выполнить действия:
- $$\frac{a^2+ab}{a^2+b^2}\cdot\left(\frac{a}{a-b}-\frac{a}{a+b}\right)$$
- $$\frac{ab-b^2}{a^2+b^2}\cdot\left(\frac{a}{a+b}+\frac{b}{a-b}\right)$$
- $$\left(\frac{c+d}{c}-\frac{2c}{c-d}\right)\cdot\frac{d-c}{c^2+d^2}$$
- $$\left(\frac{2c}{c+d}+\frac{d-c}{c}\right)\cdot\frac{c+d}{c^2+d^2}$$
$$7(y-3)-a(3-y)=7(y-3)+a(y-3)=(y-3)(7+a).$$
$$6(a-2)+a(2-a)=6(a-2)-a(a-2)=(a-2)(6-a).$$
$$b^2(a-1)-c(1-a)=b^2(a-1)+c(a-1)=(a-1)(b^2+c).$$
$$a^2(m-2)+b(2-m)=a^2(m-2)-b(m-2)=(m-2)(a^2-b).$$
$$\frac{a^2+ab}{a^2+b^2}\left(\frac{a}{a-b}-\frac{a}{a+b}\right) =\frac{a^2+ab}{a^2+b^2}\cdot \frac{a(a+b)-a(a-b)}{(a-b)(a+b)}$$
$$=\frac{a^2+ab}{a^2+b^2}\cdot \frac{2ab}{a^2-b^2} =\frac{a(a+b)}{a^2+b^2}\cdot \frac{2ab}{(a-b)(a+b)}$$
$$=\frac{2a^2b}{(a^2+b^2)(a-b)}.$$
$$\frac{ab-b^2}{a^2+b^2}\left(\frac{a}{a+b}+\frac{b}{a-b}\right) =\frac{b(a-b)}{a^2+b^2}\cdot \frac{a(a-b)+b(a+b)}{(a-b)(a+b)}$$
$$=\frac{b}{a^2+b^2}\cdot \frac{a^2-ab+ab+b^2}{a+b} =\frac{b}{a^2+b^2}\cdot \frac{a^2+b^2}{a+b} =\frac{b}{a+b}.$$
$$\left(\frac{c+d}{c}-\frac{2c}{c-d}\right)\cdot \frac{d-c}{c^2+d^2} =\frac{(c+d)(c-d)-2c^2}{c(c-d)}\cdot \frac{d-c}{c^2+d^2}$$
$$=\frac{c^2-d^2-2c^2}{c(c-d)}\cdot \frac{d-c}{c^2+d^2} =\frac{-(c^2+d^2)}{c(c-d)}\cdot \frac{d-c}{c^2+d^2} =\frac{1}{c}.$$
$$\left(\frac{2c}{c+d}+\frac{d-c}{c}\right)\cdot \frac{c+d}{c^2+d^2} =\frac{2c^2+c(d-c)}{c(c+d)}\cdot \frac{c+d}{c^2+d^2}$$
$$=\frac{c^2+cd}{c(c+d)}\cdot \frac{c+d}{c^2+d^2} =\frac{c(c+d)}{c(c+d)}\cdot \frac{1}{c^2+d^2} =\frac{1}{c}.$$
Ответ
1) $$(y-3)(7+a);$$
2) $$(a-2)(6-a);$$
3) $$(a-1)(b^2+c);$$
4) $$(m-2)(a^2-b);$$
5) $$\frac{2a^2b}{(a^2+b^2)(a-b)};$$
6) $$\frac{b}{a+b};$$
7) $$\frac{1}{c};$$
8) $$\frac{1}{c}.$$
