1089 Вправа 1089Алгебра 8 клас Бевз 2025 · § 21 · стор. 205

1089. Обчисли значення дробу. а) (x² − 3x + 2)/(x² − x − 2) при x = −1,1; x = 9; x = 11; x = 99; б) (a³ − 5a² + 6a)/(a³ − 6a² + 9a) при a = −2; a = 3,5; a = 13.

Спочатку скорочуємо дріб, потім підставляємо.

а) ${\displaystyle \frac{x^2-3x+2}{x^2-x-2}}$;

${\displaystyle x^2-3x+2}$: ${\displaystyle x_1+x_2=}$${\displaystyle 3}$;

${\displaystyle x_1{x}_2=}$${\displaystyle 2}$ → ${\displaystyle x_1=}$${\displaystyle 2,x_2=}$${\displaystyle 1}$;

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

${\displaystyle x^2-x-2}$: ${\displaystyle x_1+x_2=}$${\displaystyle 1}$;

${\displaystyle x_1{x}_2=}$${\displaystyle -2}$ → ${\displaystyle x_1=}$${\displaystyle 2,x_2=}$${\displaystyle -1}$;

${\displaystyle x^2-x-2=}$${\displaystyle \left(x-2\right)\left(x+1\right)}$;

${\displaystyle \frac{(x-1)\cancel{\left(x-2\right)}}{\cancel{\left(x-2\right)}(x+1)}=}$${\displaystyle \frac{x-1}{x+1}}$;

${\displaystyle x=}$${\displaystyle -1,1}$: ${\displaystyle \frac{-2,1}{-0,1}=}$${\displaystyle 21}$;

${\displaystyle x=}$${\displaystyle 9}$: ${\displaystyle \frac{8}{10}=}$${\displaystyle 0,8}$; ${\displaystyle x=}$${\displaystyle 11}$: ${\displaystyle \frac{10}{12}=}$${\displaystyle \frac{5}{6}}$; ${\displaystyle x=}$${\displaystyle 99}$: ${\displaystyle \frac{98}{100}=}$${\displaystyle 0,98}$;

б) ${\displaystyle \frac{a^3-5a^2+6a}{a^3-6a^2+9a}=}$${\displaystyle \frac{a(a^2-5a+6)}{a(a^2-6a+9)}}$;

${\displaystyle a^2-5a+6}$: ${\displaystyle a_1+a_2=}$${\displaystyle 5}$;

${\displaystyle a_1{a}_2=}$${\displaystyle 6}$ → ${\displaystyle a_1=}$${\displaystyle 3,a_2=}$${\displaystyle 2}$;

${\displaystyle a^2-5a+6=}$${\displaystyle \left(a-3\right)\left(a-2\right)}$;

${\displaystyle a^2-6a+9=}$${\displaystyle {(a-3)}^2}$;

${\displaystyle \frac{\cancel{a}(a-2)\cancel{\left(a-3\right)}}{\cancel{a}(a-3)\cancel{\left(a-3\right)}}=}$${\displaystyle \frac{a-2}{a-3}}$;

${\displaystyle a=}$${\displaystyle -2}$: ${\displaystyle \frac{-4}{-5}=}$${\displaystyle 0,8}$; ${\displaystyle a=}$${\displaystyle 3,5}$: ${\displaystyle \frac{1,5}{0,5}=}$${\displaystyle 3}$; ${\displaystyle a=}$${\displaystyle 13}$: ${\displaystyle \frac{11}{10}=}$${\displaystyle 1,1}$.