958 Вправа 958Алгебра 8 клас Бевз 2025 · § 19 · стор. 183

958. Користуючись формулою коренів, розв’яжи рівняння. а) 3x² − 2x − 5 = 0; б) 2z² − 3z − 2 = 0; в) 2y² − 7y + 3 = 0; г) 5z² − 8z + 3 = 0.

а) ${\displaystyle 3x^2-2x-5=}$${\displaystyle 0}$;

${\displaystyle a=}$${\displaystyle 3,b=}$${\displaystyle -2,c=}$${\displaystyle -5}$;

${\displaystyle D=}$${\displaystyle b^2-4ac=}$${\displaystyle {(-2)}^2-4\cdot 3\cdot (-5)=}$${\displaystyle 4+60=}$${\displaystyle 64}$;

${\displaystyle D>0}$, два корені:

${\displaystyle x_{1,2}=}$${\displaystyle \frac{-b\pm \sqrt{D}}{2a}=}$${\displaystyle \frac{2\pm \sqrt{64}}{6}=}$${\displaystyle \frac{2\pm 8}{6}}$;

${\displaystyle x_1=}$${\displaystyle \frac{2+8}{6}=}$${\displaystyle \frac{5}{3}}$;

${\displaystyle x_2=}$${\displaystyle \frac{2-8}{6}=}$${\displaystyle -1}$;

б) ${\displaystyle 2z^2-3z-2=}$${\displaystyle 0}$;

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

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

${\displaystyle D>0}$, два корені:

${\displaystyle z_{1,2}=}$${\displaystyle \frac{-b\pm \sqrt{D}}{2a}=}$${\displaystyle \frac{3\pm \sqrt{25}}{4}=}$${\displaystyle \frac{3\pm 5}{4}}$;

${\displaystyle z_1=}$${\displaystyle \frac{3+5}{4}=}$${\displaystyle 2}$;

${\displaystyle z_2=}$${\displaystyle \frac{3-5}{4}=}$${\displaystyle -0,5}$;

в) ${\displaystyle 2y^2-7y+3=}$${\displaystyle 0}$;

${\displaystyle a=}$${\displaystyle 2,b=}$${\displaystyle -7,c=}$${\displaystyle 3}$;

${\displaystyle D=}$${\displaystyle b^2-4ac=}$${\displaystyle {(-7)}^2-4\cdot 2\cdot 3=}$${\displaystyle 49-24=}$${\displaystyle 25}$;

${\displaystyle D>0}$, два корені:

${\displaystyle y_{1,2}=}$${\displaystyle \frac{-b\pm \sqrt{D}}{2a}=}$${\displaystyle \frac{7\pm \sqrt{25}}{4}=}$${\displaystyle \frac{7\pm 5}{4}}$;

${\displaystyle y_1=}$${\displaystyle \frac{7+5}{4}=}$${\displaystyle 3}$;

${\displaystyle y_2=}$${\displaystyle \frac{7-5}{4}=}$${\displaystyle 0,5}$;

г) ${\displaystyle 5z^2-8z+3=}$${\displaystyle 0}$;

${\displaystyle a=}$${\displaystyle 5,b=}$${\displaystyle -8,c=}$${\displaystyle 3}$;

${\displaystyle D=}$${\displaystyle b^2-4ac=}$${\displaystyle {(-8)}^2-4\cdot 5\cdot 3=}$${\displaystyle 64-60=}$${\displaystyle 4}$;

${\displaystyle D>0}$, два корені:

${\displaystyle z_{1,2}=}$${\displaystyle \frac{-b\pm \sqrt{D}}{2a}=}$${\displaystyle \frac{8\pm \sqrt{4}}{10}=}$${\displaystyle \frac{8\pm 2}{10}}$;

${\displaystyle z_1=}$${\displaystyle \frac{8+2}{10}=}$${\displaystyle 1}$;

${\displaystyle z_2=}$${\displaystyle \frac{8-2}{10}=}$${\displaystyle 0,6}$.