848° Вправа 848Алгебра 8 клас Тарасенкова 2025 · § 21 · стор. 255

848. Розв’яжіть рівняння:

1) ${\displaystyle x^4-10x^2+9=}$${\displaystyle 0}$;

${\displaystyle t=}$${\displaystyle x^2}$, ${\displaystyle t\ge 0}$;

${\displaystyle t^2-10t+9=}$${\displaystyle 0}$;

${\displaystyle t_1+t_2=}$${\displaystyle 10}$; ${\displaystyle t_1\cdot t_2=}$${\displaystyle 9}$;

${\displaystyle t_1=}$${\displaystyle 1}$; ${\displaystyle t_2=}$${\displaystyle 9}$;

${\displaystyle x^2=}$${\displaystyle 1}$ або ${\displaystyle x^2=}$${\displaystyle 9}$;

${\displaystyle x_{1,2}=}$${\displaystyle \pm 1}$; ${\displaystyle x_{3,4}=}$${\displaystyle \pm 3}$;

2) ${\displaystyle x^4-x^2-12=}$${\displaystyle 0}$;

${\displaystyle t=}$${\displaystyle x^2}$, ${\displaystyle t\ge 0}$;

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

${\displaystyle t_1+t_2=}$${\displaystyle 1}$; ${\displaystyle t_1\cdot t_2=}$${\displaystyle -12}$;

${\displaystyle t_1=}$${\displaystyle 4}$;

${\displaystyle t_2=}$${\displaystyle -3}$ — не задовольняє ${\displaystyle t\ge 0}$;

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

${\displaystyle x_{1,2}=}$${\displaystyle \pm 2}$;

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

${\displaystyle t=}$${\displaystyle x^2}$, ${\displaystyle t\ge 0}$;

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

${\displaystyle t_1+t_2=}$${\displaystyle 3}$; ${\displaystyle t_1\cdot t_2=}$${\displaystyle -4}$;

${\displaystyle t_1=}$${\displaystyle 4}$;

${\displaystyle t_2=}$${\displaystyle -1}$ — не задовольняє ${\displaystyle t\ge 0}$;

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

${\displaystyle x_{1,2}=}$${\displaystyle \pm 2}$;

4) ${\displaystyle x^4-8x^2-9=}$${\displaystyle 0}$;

${\displaystyle t=}$${\displaystyle x^2}$, ${\displaystyle t\ge 0}$;

${\displaystyle t^2-8t-9=}$${\displaystyle 0}$;

${\displaystyle t_1+t_2=}$${\displaystyle 8}$; ${\displaystyle t_1\cdot t_2=}$${\displaystyle -9}$;

${\displaystyle t_1=}$${\displaystyle 9}$;

${\displaystyle t_2=}$${\displaystyle -1}$ — не задовольняє ${\displaystyle t\ge 0}$;

${\displaystyle x^2=}$${\displaystyle 9}$;

${\displaystyle x_{1,2}=}$${\displaystyle \pm 3}$;

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

${\displaystyle t=}$${\displaystyle x^2}$, ${\displaystyle t\ge 0}$;

${\displaystyle t^2-5t+6=}$${\displaystyle 0}$;

${\displaystyle t_1+t_2=}$${\displaystyle 5}$; ${\displaystyle t_1\cdot t_2=}$${\displaystyle 6}$;

${\displaystyle t_1=}$${\displaystyle 2}$; ${\displaystyle t_2=}$${\displaystyle 3}$;

${\displaystyle x^2=}$${\displaystyle 2}$ або ${\displaystyle x^2=}$${\displaystyle 3}$;

${\displaystyle x_{1,2}=}$${\displaystyle \pm \sqrt{2}}$; ${\displaystyle x_{3,4}=}$${\displaystyle \pm \sqrt{3}}$;

6) ${\displaystyle x^4+2x^2-8=}$${\displaystyle 0}$;

${\displaystyle t=}$${\displaystyle x^2}$, ${\displaystyle t\ge 0}$;

${\displaystyle t^2+2t-8=}$${\displaystyle 0}$;

${\displaystyle t_1+t_2=}$${\displaystyle -2}$; ${\displaystyle t_1\cdot t_2=}$${\displaystyle -8}$;

${\displaystyle t_1=}$${\displaystyle 2}$;

${\displaystyle t_2=}$${\displaystyle -4}$ — не задовольняє ${\displaystyle t\ge 0}$;

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

${\displaystyle x_{1,2}=}$${\displaystyle \pm \sqrt{2}}$;

7) ${\displaystyle x^4-20x^2+64=}$${\displaystyle 0}$;

${\displaystyle t=}$${\displaystyle x^2}$, ${\displaystyle t\ge 0}$;

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

${\displaystyle t_1+t_2=}$${\displaystyle 20}$; ${\displaystyle t_1\cdot t_2=}$${\displaystyle 64}$;

${\displaystyle t_1=}$${\displaystyle 4}$; ${\displaystyle t_2=}$${\displaystyle 16}$;

${\displaystyle x^2=}$${\displaystyle 4}$ або ${\displaystyle x^2=}$${\displaystyle 16}$;

${\displaystyle x_{1,2}=}$${\displaystyle \pm 2}$; ${\displaystyle x_{3,4}=}$${\displaystyle \pm 4}$;

8) ${\displaystyle x^4+5x^2+4=}$${\displaystyle 0}$;

${\displaystyle t=}$${\displaystyle x^2}$, ${\displaystyle t\ge 0}$;

${\displaystyle t^2+5t+4=}$${\displaystyle 0}$;

${\displaystyle t_1+t_2=}$${\displaystyle -5}$; ${\displaystyle t_1\cdot t_2=}$${\displaystyle 4}$;

${\displaystyle t_1=}$${\displaystyle -1}$; ${\displaystyle t_2=}$${\displaystyle -4}$;

${\displaystyle t_1<0}$, ${\displaystyle t_2<0}$ — коренів немає.