845° Вправа 845Алгебра 8 клас Тарасенкова 2025 · § 21 · стор. 254

845. Розв’яжіть рівняння:

1) ${\displaystyle \frac{x+4}{x-2}=}$${\displaystyle x}$;

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

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

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

${\displaystyle \{\begin{array}{rl}& -x^2+3x+4=0,\\ & x-2\ne 0\end{array}}$;

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

${\displaystyle x_1+x_2=}$${\displaystyle 3}$; ${\displaystyle x_1\cdot x_2=}$${\displaystyle -4}$;

${\displaystyle x_1=}$${\displaystyle 4}$; ${\displaystyle x_2=}$${\displaystyle -1}$;

${\displaystyle x\ne 2}$;

${\displaystyle x_1=}$${\displaystyle 4}$; ${\displaystyle x_2=}$${\displaystyle -1}$;

2) ${\displaystyle \frac{32}{x}+\frac{x}{2}=}$${\displaystyle 10}$;

${\displaystyle \frac{32\cdot 2+x^2-10\cdot 2x}{2x}=}$${\displaystyle 0}$;

${\displaystyle \frac{64+x^2-20x}{2x}=}$${\displaystyle 0}$;

${\displaystyle \{\begin{array}{rl}& x^2-20x+64=0,\\ & 2x\ne 0\end{array}}$;

${\displaystyle D=}$${\displaystyle {(-20)}^2-4\cdot 1\cdot 64=}$${\displaystyle 400-256=}$${\displaystyle 144}$;

${\displaystyle x_{1,2}=}$${\displaystyle \frac{20\pm \sqrt{144}}{2\cdot 1}=}$${\displaystyle \frac{20\pm 12}{2}}$;

${\displaystyle x_1=}$${\displaystyle \frac{20+12}{2}=}$${\displaystyle 16}$;

${\displaystyle x_2=}$${\displaystyle \frac{20-12}{2}=}$${\displaystyle 4}$;

${\displaystyle x\ne 0}$;

${\displaystyle x_1=}$${\displaystyle 16}$; ${\displaystyle x_2=}$${\displaystyle 4}$.

3) ${\displaystyle \frac{2x^2+5x+2}{x^2-4}=}$${\displaystyle 1}$;

${\displaystyle \frac{2x^2+5x+2-(x^2-4)}{x^2-4}=}$${\displaystyle 0}$;

${\displaystyle \frac{2x^2+5x+2-x^2+4}{x^2-4}=}$${\displaystyle 0}$;

${\displaystyle \frac{x^2+5x+6}{x^2-4}=}$${\displaystyle 0}$;

${\displaystyle \{\begin{array}{rl}& x^2+5x+6=0,\\ & x^2-4\ne 0\end{array}}$;

${\displaystyle x_1+x_2=}$${\displaystyle -5}$; ${\displaystyle x_1\cdot x_2=}$${\displaystyle 6}$;

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

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

${\displaystyle x^2\ne 4}$;

${\displaystyle x\ne 2}$, ${\displaystyle x\ne -2}$;

${\displaystyle x=}$${\displaystyle -3}$.