1389 Вправа 1389Алгебра 8 клас Бевз 2025 · Повторення · стор. 264

1389. Розв’яжи рівняння. а) x − 6√x + 5 = 0; б) 2x − 3√x − 9 = 0.

а) ${\displaystyle x-6\sqrt{x}+5=}$${\displaystyle 0}$;

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

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

${\displaystyle a=}$${\displaystyle 1,b=}$${\displaystyle -6,c=}$${\displaystyle 5}$;

${\displaystyle D=}$${\displaystyle b^2-4ac=}$${\displaystyle {(-6)}^2-4\cdot 1\cdot 5=}$${\displaystyle 36-20=}$${\displaystyle 16}$;

${\displaystyle D>0}$, два корені:

${\displaystyle t_{1,2}=}$${\displaystyle \frac{-b\pm \sqrt{D}}{2a}=}$${\displaystyle \frac{6\pm \sqrt{16}}{2}=}$${\displaystyle \frac{6\pm 4}{2}}$;

${\displaystyle t_1=}$${\displaystyle \frac{6+4}{2}=}$${\displaystyle 5}$;

${\displaystyle t_2=}$${\displaystyle \frac{6-4}{2}=}$${\displaystyle 1}$;

${\displaystyle \sqrt{x}=}$${\displaystyle 5}$;

${\displaystyle x=}$${\displaystyle 25}$;

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

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

б) ${\displaystyle 2x-3\sqrt{x}-9=}$${\displaystyle 0}$;

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

${\displaystyle 2t^2-3t-9=}$${\displaystyle 0}$;

${\displaystyle a=}$${\displaystyle 2,b=}$${\displaystyle -3,c=}$${\displaystyle -9}$;

${\displaystyle D=}$${\displaystyle b^2-4ac=}$${\displaystyle {(-3)}^2-4\cdot 2\cdot (-9)=}$${\displaystyle 9+72=}$${\displaystyle 81}$;

${\displaystyle D>0}$, два корені:

${\displaystyle t_{1,2}=}$${\displaystyle \frac{-b\pm \sqrt{D}}{2a}=}$${\displaystyle \frac{3\pm \sqrt{81}}{4}=}$${\displaystyle \frac{3\pm 9}{4}}$;

${\displaystyle t_1=}$${\displaystyle \frac{3+9}{4}=}$${\displaystyle 3}$;

${\displaystyle t_2=}$${\displaystyle \frac{3-9}{4}=}$${\displaystyle -1,5}$;

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

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

${\displaystyle t=}$${\displaystyle -1,5<0}$ — не підходить.