1391 Вправа 1391Алгебра 8 клас Бевз 2025 · Повторення · стор. 264

1391. Розв’яжи рівняння. а) x + 2 − 13√(x + 2) + 42 = 0; б) x − 3 + 4√(x − 3) − 12 = 0.

а) ${\displaystyle x+2-13\sqrt{x+2}+42=}$${\displaystyle 0}$;

заміна ${\displaystyle \sqrt{x+2}=}$${\displaystyle t}$, ${\displaystyle t\ge 0}$:

${\displaystyle t^2-13t+42=}$${\displaystyle 0}$;

${\displaystyle a=}$${\displaystyle 1,b=}$${\displaystyle -13,c=}$${\displaystyle 42}$;

${\displaystyle D=}$${\displaystyle b^2-4ac=}$${\displaystyle {(-13)}^2-4\cdot 1\cdot 42=}$${\displaystyle 169-168=}$${\displaystyle 1}$;

${\displaystyle D>0}$, два корені:

${\displaystyle t_{1,2}=}$${\displaystyle \frac{-b\pm \sqrt{D}}{2a}=}$${\displaystyle \frac{13\pm \sqrt{1}}{2}=}$${\displaystyle \frac{13\pm 1}{2}}$;

${\displaystyle t_1=}$${\displaystyle \frac{13+1}{2}=}$${\displaystyle 7}$;

${\displaystyle t_2=}$${\displaystyle \frac{13-1}{2}=}$${\displaystyle 6}$;

${\displaystyle \sqrt{x+2}=}$${\displaystyle 7}$;

${\displaystyle x+2=}$${\displaystyle 49}$;

${\displaystyle x=}$${\displaystyle 47}$;

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

${\displaystyle x+2=}$${\displaystyle 36}$;

${\displaystyle x=}$${\displaystyle 34}$;

б) ${\displaystyle x-3+4\sqrt{x-3}-12=}$${\displaystyle 0}$;

заміна ${\displaystyle \sqrt{x-3}=}$${\displaystyle t}$, ${\displaystyle t\ge 0}$:

${\displaystyle t^2+4t-12=}$${\displaystyle 0}$;

${\displaystyle a=}$${\displaystyle 1,b=}$${\displaystyle 4,c=}$${\displaystyle -12}$;

${\displaystyle D=}$${\displaystyle b^2-4ac=}$${\displaystyle 4^2-4\cdot 1\cdot (-12)=}$${\displaystyle 16+48=}$${\displaystyle 64}$;

${\displaystyle D>0}$, два корені:

${\displaystyle t_{1,2}=}$${\displaystyle \frac{-b\pm \sqrt{D}}{2a}=}$${\displaystyle \frac{-4\pm \sqrt{64}}{2}=}$${\displaystyle \frac{-4\pm 8}{2}}$;

${\displaystyle t_1=}$${\displaystyle \frac{-4+8}{2}=}$${\displaystyle 2}$;

${\displaystyle t_2=}$${\displaystyle \frac{-4-8}{2}=}$${\displaystyle -6}$;

${\displaystyle \sqrt{x-3}=}$${\displaystyle 2}$;

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

${\displaystyle x=}$${\displaystyle 7}$;

${\displaystyle t=}$${\displaystyle -6<0}$ — не підходить.