957 Вправа 957Алгебра 8 клас Бевз 2025 · § 19 · стор. 183

957. Користуючись формулою коренів, розв’яжи рівняння. а) x² + 3x − 4 = 0; б) x² − 7x − 30 = 0; в) x² − x − 12 = 0; г) x² + x − 20 = 0.

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

${\displaystyle a=}$${\displaystyle 1,b=}$${\displaystyle 3,c=}$${\displaystyle -4}$;

${\displaystyle D=}$${\displaystyle b^2-4ac=}$${\displaystyle 3^2-4\cdot 1\cdot (-4)=}$${\displaystyle 9+16=}$${\displaystyle 25}$;

${\displaystyle D>0}$, два корені:

${\displaystyle x_{1,2}=}$${\displaystyle \frac{-b\pm \sqrt{D}}{2a}=}$${\displaystyle \frac{-3\pm \sqrt{25}}{2}=}$${\displaystyle \frac{-3\pm 5}{2}}$;

${\displaystyle x_1=}$${\displaystyle \frac{-3+5}{2}=}$${\displaystyle 1}$;

${\displaystyle x_2=}$${\displaystyle \frac{-3-5}{2}=}$${\displaystyle -4}$;

б) ${\displaystyle x^2-7x-30=}$${\displaystyle 0}$;

${\displaystyle a=}$${\displaystyle 1,b=}$${\displaystyle -7,c=}$${\displaystyle -30}$;

${\displaystyle D=}$${\displaystyle b^2-4ac=}$${\displaystyle {(-7)}^2-4\cdot 1\cdot (-30)=}$${\displaystyle 49+120=}$${\displaystyle 169}$;

${\displaystyle D>0}$, два корені:

${\displaystyle x_{1,2}=}$${\displaystyle \frac{-b\pm \sqrt{D}}{2a}=}$${\displaystyle \frac{7\pm \sqrt{169}}{2}=}$${\displaystyle \frac{7\pm 13}{2}}$;

${\displaystyle x_1=}$${\displaystyle \frac{7+13}{2}=}$${\displaystyle 10}$;

${\displaystyle x_2=}$${\displaystyle \frac{7-13}{2}=}$${\displaystyle -3}$;

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

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

${\displaystyle D=}$${\displaystyle b^2-4ac=}$${\displaystyle {(-1)}^2-4\cdot 1\cdot (-12)=}$${\displaystyle 1+48=}$${\displaystyle 49}$;

${\displaystyle D>0}$, два корені:

${\displaystyle x_{1,2}=}$${\displaystyle \frac{-b\pm \sqrt{D}}{2a}=}$${\displaystyle \frac{1\pm \sqrt{49}}{2}=}$${\displaystyle \frac{1\pm 7}{2}}$;

${\displaystyle x_1=}$${\displaystyle \frac{1+7}{2}=}$${\displaystyle 4}$;

${\displaystyle x_2=}$${\displaystyle \frac{1-7}{2}=}$${\displaystyle -3}$;

г) ${\displaystyle x^2+x-20=}$${\displaystyle 0}$;

${\displaystyle a=}$${\displaystyle 1,b=}$${\displaystyle 1,c=}$${\displaystyle -20}$;

${\displaystyle D=}$${\displaystyle b^2-4ac=}$${\displaystyle 1^2-4\cdot 1\cdot (-20)=}$${\displaystyle 1+80=}$${\displaystyle 81}$;

${\displaystyle D>0}$, два корені:

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

${\displaystyle x_1=}$${\displaystyle \frac{-1+9}{2}=}$${\displaystyle 4}$;

${\displaystyle x_2=}$${\displaystyle \frac{-1-9}{2}=}$${\displaystyle -5}$.