858 Вправа 858Алгебра 8 клас Тарасенкова 2025 · § 21 · стор. 256

858. Розв’яжіть рівняння:

1) ${\displaystyle \frac{3x-2}{x}-\frac{1}{x-2}=}$${\displaystyle \frac{3x+4}{x^2-2x}}$; ОДЗ: ${\displaystyle x\ne 0}$, ${\displaystyle x\ne 2}$;

${\displaystyle \frac{{3x-2}^{\mathrm{\setminus }x-2}}{x}-\frac{1^{\mathrm{\setminus }x}}{x-2}-\frac{3x+4}{x(x-2)}=}$${\displaystyle 0}$;

${\displaystyle \frac{(3x-2)(x-2)-1\cdot x-(3x+4)}{x(x-2)}=}$${\displaystyle 0}$;

${\displaystyle 3x^2-6x-2x+4-x-3x-4=}$${\displaystyle 0}$;

${\displaystyle 3x^2-12x=}$${\displaystyle 0}$;

${\displaystyle 3x(x-4)=}$${\displaystyle 0}$;

${\displaystyle 3x=}$${\displaystyle 0}$ або ${\displaystyle x-4=}$${\displaystyle 0}$;

${\displaystyle x=}$${\displaystyle 0}$ — не задовольняє ОДЗ;

${\displaystyle x=}$${\displaystyle 4}$;

2) ${\displaystyle \frac{3}{x+1}-\frac{19}{x-7}=}$${\displaystyle 8}$; ОДЗ: ${\displaystyle x\ne -1}$, ${\displaystyle x\ne 7}$;

${\displaystyle \frac{3(x-7)-19(x+1)-8(x+1)(x-7)}{(x+1)(x-7)}=}$${\displaystyle 0}$;

${\displaystyle 3x-21-19x-19-8(x^2-6x-7)=}$${\displaystyle 0}$;

${\displaystyle -16x-40-8x^2+48x+56=}$${\displaystyle 0}$;

${\displaystyle -8x^2+32x+16=}$${\displaystyle 0}$ | ${\displaystyle :\left(-8\right)}$

${\displaystyle x^2-4x-2=}$${\displaystyle 0}$;

${\displaystyle D=}$${\displaystyle {(-4)}^2-4\cdot 1\cdot (-2)=}$${\displaystyle 16+8=}$${\displaystyle 24}$;

${\displaystyle x_{1,2}=}$${\displaystyle \frac{4\pm \sqrt{24}}{2\cdot 1}=}$${\displaystyle \frac{4\pm 2\sqrt{6}}{2}=}$${\displaystyle 2\pm \sqrt{6}}$;

${\displaystyle x_1=}$${\displaystyle 2+\sqrt{6}}$;

${\displaystyle x_2=}$${\displaystyle 2-\sqrt{6}}$;

3) ${\displaystyle \frac{4}{x+3}+\frac{x-5}{x-3}=}$${\displaystyle -\frac{24}{x^2-9}}$; ОДЗ: ${\displaystyle x\ne -3}$, ${\displaystyle x\ne 3}$;

${\displaystyle \frac{4(x-3)+(x-5)(x+3)+24}{(x-3)(x+3)}=}$${\displaystyle 0}$;

${\displaystyle 4x-12+x^2+3x-5x-15+24=}$${\displaystyle 0}$;

${\displaystyle x^2+2x-3=}$${\displaystyle 0}$;

${\displaystyle x_1+x_2=}$${\displaystyle -2}$; ${\displaystyle x_1\cdot x_2=}$${\displaystyle -3}$;

${\displaystyle x_1=}$${\displaystyle -3}$; ${\displaystyle x_2=}$${\displaystyle 1}$;

${\displaystyle x_1=}$${\displaystyle -3}$ — не задовольняє ОДЗ;

${\displaystyle x=}$${\displaystyle 1}$;

4) ${\displaystyle \frac{2}{x+7}+\frac{7-x}{x^2+7x}=}$${\displaystyle 1}$; ОДЗ: ${\displaystyle x\ne 0}$, ${\displaystyle x\ne -7}$;

${\displaystyle \frac{2\cdot x+(7-x)-1\cdot x(x+7)}{x(x+7)}=}$${\displaystyle 0}$;

${\displaystyle 2x+7-x-x^2-7x=}$${\displaystyle 0}$;

${\displaystyle -x^2-6x+7=}$${\displaystyle 0}$;

${\displaystyle x^2+6x-7=}$${\displaystyle 0}$;

${\displaystyle x_1+x_2=}$${\displaystyle -6}$; ${\displaystyle x_1\cdot x_2=}$${\displaystyle -7}$;

${\displaystyle x_1=}$${\displaystyle -7}$; ${\displaystyle x_2=}$${\displaystyle 1}$;

${\displaystyle x_1=}$${\displaystyle -7}$ — не задовольняє ОДЗ;

${\displaystyle x=}$${\displaystyle 1}$;

5) ${\displaystyle \frac{x+7}{x+4}+\frac{1}{x^2+7x+12}=}$${\displaystyle \frac{1}{x+3}}$; ${\displaystyle x^2+7x+12=}$${\displaystyle (x+3)(x+4)}$; ОДЗ: ${\displaystyle x\ne -4}$, ${\displaystyle x\ne -3}$;

${\displaystyle \frac{(x+7)(x+3)+1-1\cdot (x+4)}{(x+3)(x+4)}=}$${\displaystyle 0}$;

${\displaystyle x^2+3x+7x+21+1-x-4=}$${\displaystyle 0}$;

${\displaystyle x^2+9x+18=}$${\displaystyle 0}$;

${\displaystyle x_1+x_2=}$${\displaystyle -9}$; ${\displaystyle x_1\cdot x_2=}$${\displaystyle 18}$;

${\displaystyle x_1=}$${\displaystyle -6}$; ${\displaystyle x_2=}$${\displaystyle -3}$;

${\displaystyle x_2=}$${\displaystyle -3}$ — не задовольняє ОДЗ;

${\displaystyle x=}$${\displaystyle -6}$;

6) ${\displaystyle \frac{1}{x^2-3x+2}+\frac{3}{x-1}=}$${\displaystyle \frac{3-x}{x-2}}$; ${\displaystyle x^2-3x+2=}$${\displaystyle (x-1)(x-2)}$; ОДЗ: ${\displaystyle x\ne 1}$, ${\displaystyle x\ne 2}$;

${\displaystyle \frac{1+3(x-2)-(3-x)(x-1)}{(x-1)(x-2)}=}$${\displaystyle 0}$;

${\displaystyle 1+3x-6-(3x-3-x^2+x)=}$${\displaystyle 0}$;

${\displaystyle 3x-5-3x+3+x^2-x=}$${\displaystyle 0}$;

${\displaystyle x^2-x-2=}$${\displaystyle 0}$;

${\displaystyle x_1+x_2=}$${\displaystyle 1}$; ${\displaystyle x_1\cdot x_2=}$${\displaystyle -2}$;

${\displaystyle x_1=}$${\displaystyle 2}$; ${\displaystyle x_2=}$${\displaystyle -1}$;

${\displaystyle x_1=}$${\displaystyle 2}$ — не задовольняє ОДЗ;

${\displaystyle x=}$${\displaystyle -1}$.