956 Вправа 956Алгебра 8 клас Бевз 2025 · § 19 · стор. 183

956. Користуючись формулою коренів, розв’яжи рівняння. а) x² − 7x − 18 = 0; б) x² + 7x − 18 = 0; в) x² + x − 6 = 0; г) x² − x − 42 = 0.

Формула коренів: ${\displaystyle x_{1,2}=}$${\displaystyle \frac{-b\pm \sqrt{D}}{2a}}$, де ${\displaystyle D=}$${\displaystyle b^2-4ac}$.

а) ${\displaystyle x^2-7x-18=}$${\displaystyle 0}$;

${\displaystyle a=}$${\displaystyle 1,b=}$${\displaystyle -7,c=}$${\displaystyle -18}$;

${\displaystyle D=}$${\displaystyle b^2-4ac=}$${\displaystyle {(-7)}^2-4\cdot 1\cdot (-18)=}$${\displaystyle 49+72=}$${\displaystyle 121}$;

${\displaystyle D>0}$, два корені:

${\displaystyle x_{1,2}=}$${\displaystyle \frac{-b\pm \sqrt{D}}{2a}=}$${\displaystyle \frac{7\pm \sqrt{121}}{2}=}$${\displaystyle \frac{7\pm 11}{2}}$;

${\displaystyle x_1=}$${\displaystyle \frac{7+11}{2}=}$${\displaystyle 9}$;

${\displaystyle x_2=}$${\displaystyle \frac{7-11}{2}=}$${\displaystyle -2}$;

б) ${\displaystyle x^2+7x-18=}$${\displaystyle 0}$;

${\displaystyle a=}$${\displaystyle 1,b=}$${\displaystyle 7,c=}$${\displaystyle -18}$;

${\displaystyle D=}$${\displaystyle b^2-4ac=}$${\displaystyle 7^2-4\cdot 1\cdot (-18)=}$${\displaystyle 49+72=}$${\displaystyle 121}$;

${\displaystyle D>0}$, два корені:

${\displaystyle x_{1,2}=}$${\displaystyle \frac{-b\pm \sqrt{D}}{2a}=}$${\displaystyle \frac{-7\pm \sqrt{121}}{2}=}$${\displaystyle \frac{-7\pm 11}{2}}$;

${\displaystyle x_1=}$${\displaystyle \frac{-7+11}{2}=}$${\displaystyle 2}$;

${\displaystyle x_2=}$${\displaystyle \frac{-7-11}{2}=}$${\displaystyle -9}$;

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

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

${\displaystyle D=}$${\displaystyle b^2-4ac=}$${\displaystyle 1^2-4\cdot 1\cdot (-6)=}$${\displaystyle 1+24=}$${\displaystyle 25}$;

${\displaystyle D>0}$, два корені:

${\displaystyle x_{1,2}=}$${\displaystyle \frac{-b\pm \sqrt{D}}{2a}=}$${\displaystyle \frac{-1\pm \sqrt{25}}{2}=}$${\displaystyle \frac{-1\pm 5}{2}}$;

${\displaystyle x_1=}$${\displaystyle \frac{-1+5}{2}=}$${\displaystyle 2}$;

${\displaystyle x_2=}$${\displaystyle \frac{-1-5}{2}=}$${\displaystyle -3}$;

г) ${\displaystyle x^2-x-42=}$${\displaystyle 0}$;

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

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

${\displaystyle D>0}$, два корені:

${\displaystyle x_{1,2}=}$${\displaystyle \frac{-b\pm \sqrt{D}}{2a}=}$${\displaystyle \frac{1\pm \sqrt{169}}{2}=}$${\displaystyle \frac{1\pm 13}{2}}$;

${\displaystyle x_1=}$${\displaystyle \frac{1+13}{2}=}$${\displaystyle 7}$;

${\displaystyle x_2=}$${\displaystyle \frac{1-13}{2}=}$${\displaystyle -6}$.