Упр.701 ГДЗ Колягин Ткачёва 8 класс (Алгебра)

1. $$\frac{3}{x+2}=4+\frac{3}{x-1}$$
   
   $$\frac{3}{x+2}-4-\frac{3}{x-1}=0$$
   
   $$\frac{3(x-1)-4(x+2)(x-1)-3(x+2)}{(x+2)(x-1)}=0$$
   
   $$-4x^2-4x-1=0$$
   
   $$\left(2x+1\right)^2=0$$
   
   $$x=-\frac12,\quad x\neq -2,\; x\neq 1.$$
2. $$\frac{1}{x+1}=3+\frac{3}{3x-1}$$
   
   $$\frac{1}{x+1}-3-\frac{3}{3x-1}=0$$
   
   $$\frac{1-3(x+1)(3x-1)-3(x+1)}{(x+1)(3x-1)}=0$$
   
   $$-9x^2-6x-1=0$$
   
   $$\left(3x+1\right)^2=0$$
   
   $$x=-\frac13,\quad x\neq -1,\; x\neq \frac13.$$
3. $$1+\frac{5x}{x+1}=\frac{6x+2}{(x+1)^2}$$
   
   $$\frac{(x+1)^2+5x(x+1)-(6x+2)}{(x+1)^2}=0$$
   
   $$6x^2+x-1=0,\quad x\neq -1$$
   
   $$D=1+24=25$$
   
   $$x_1=\frac{-1-5}{12}=-\frac12,\qquad x_2=\frac{-1+5}{12}=\frac13.$$
4. $$2+\frac{x}{x+2}=\frac{12-x}{(x+2)^2}$$
   
   $$\frac{2(x+2)^2+x(x+2)-(12-x)}{(x+2)^2}=0$$
   
   $$3x^2+11x-4=0,\quad x\neq -2$$
   
   $$D=121+48=169$$
   
   $$x_1=\frac{-11-13}{6}=-4,\qquad x_2=\frac{-11+13}{6}=\frac13.$$
5. $$\frac{3x}{x+2}+\frac{1}{x-2}=\frac{4}{x^2-4}$$
   
   $$\frac{3x(x-2)+(x+2)-4}{x^2-4}=0$$
   
   $$3x^2-5x-2=0,\quad x\neq \pm 2$$
   
   $$D=25+24=49$$
   
   $$x_1=\frac{5-7}{6}=-\frac13,\qquad x_2=\frac{5+7}{6}=2.$$
   
   $$x=2 \text{ не подходит},\ \text{так как } x\neq 2.$$
6. $$\frac{2x}{x-3}-\frac{1}{x+3}=\frac{6}{x^2-9}$$
   
   $$\frac{2x(x+3)-(x-3)-6}{x^2-9}=0$$
   
   $$2x^2+5x-3=0,\quad x\neq \pm 3$$
   
   $$D=25+24=49$$
   
   $$x_1=\frac{-5-7}{4}=-3,\qquad x_2=\frac{-5+7}{4}=\frac12.$$
   
   $$x=-3 \text{ не подходит},\ \text{так как } x\neq -3.$$

Ответ

1. $$x=-\frac12$$
2. $$x=-\frac13$$
3. $$x=-\frac12,\ \frac13$$
4. $$x=-4,\ \frac13$$
5. $$x=-\frac13$$
6. $$x=\frac12$$

Вычисления

1. $$\frac{27}{32}\cdot\frac{8}{162}\cdot\frac{72}{69}=\frac{1}{23}$$
2. $$\frac{38}{147}\cdot\frac{91}{152}\cdot\frac{65}{264}=\frac{845}{22176}$$
3. $$\left(\frac{5}{8}+\frac{7}{12}\right)\cdot\left(3\frac{23}{58}-2\frac{9}{58}\right)=\frac32$$
4. $$\left(\frac34+\frac29\right)\cdot\left(2\frac{23}{56}-3\frac{15}{56}\right)=-\frac56$$
5. $$34{,}17:1{,}7+\left(2\frac34+0{,}15\right):\frac45-23\frac38=\frac{7}{20}$$
6. $$5{,}86-3\frac56\cdot\frac{15}{23}+\frac{15}{28}:4\frac27=3{,}485$$
7. $$\frac{12\frac45\cdot 3\frac34-4\frac4{11}\cdot 4\frac18}{11\frac23\cdot 2\frac47}=1$$
8. $$\frac{5\frac17\cdot 5\frac14+5\frac58\cdot 3\frac15}{10\frac5{13}:1\frac1{26}}=\frac92$$

Ответ

1. $$\frac{1}{23}$$
2. $$\frac{845}{22176}$$
3. $$\frac32$$
4. $$-\frac56$$
5. $$\frac{7}{20}$$
6. $$3{,}485$$
7. $$1$$
8. $$\frac92$$