961 Вправа 961Алгебра 8 клас Бевз 2025 · § 19 · стор. 183

961. Користуючись формулою коренів, розв’яжи рівняння. а) −x² + 2x + 15 = 0; б) 9x² + 30x + 25 = 0; в) 5x² + 3x + 1 = 0; г) x² + 4x + 1 = 0; ґ) −x² + 2x + 5 = 0; д) 3x² − 2x − 3 = 0.

а) ${\displaystyle -x^2+2x+15=}$${\displaystyle 0\mid \cdot \left(-1\right)}$;

${\displaystyle x^2-2x-15=}$${\displaystyle 0}$;

${\displaystyle a=}$${\displaystyle 1,b=}$${\displaystyle -2,c=}$${\displaystyle -15}$;

${\displaystyle D=}$${\displaystyle b^2-4ac=}$${\displaystyle {(-2)}^2-4\cdot 1\cdot (-15)=}$${\displaystyle 4+60=}$${\displaystyle 64}$;

${\displaystyle D>0}$, два корені:

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

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

${\displaystyle x_2=}$${\displaystyle \frac{2-8}{2}=}$${\displaystyle -3}$;

б) ${\displaystyle 9x^2+30x+25=}$${\displaystyle 0}$;

${\displaystyle a=}$${\displaystyle 9,b=}$${\displaystyle 30,c=}$${\displaystyle 25}$;

${\displaystyle D=}$${\displaystyle b^2-4ac=}$${\displaystyle 30^2-4\cdot 9\cdot 25=}$${\displaystyle 900-900=}$${\displaystyle 0}$;

${\displaystyle D=}$${\displaystyle 0}$, один корінь:

${\displaystyle x=}$${\displaystyle -\frac{b}{2a}=}$${\displaystyle \frac{-30}{18}=}$${\displaystyle -\frac{5}{3}}$;

в) ${\displaystyle 5x^2+3x+1=}$${\displaystyle 0}$;

${\displaystyle a=}$${\displaystyle 5,b=}$${\displaystyle 3,c=}$${\displaystyle 1}$;

${\displaystyle D=}$${\displaystyle b^2-4ac=}$${\displaystyle 3^2-4\cdot 5\cdot 1=}$${\displaystyle 9-20=}$${\displaystyle -11}$;

${\displaystyle D<0}$ — коренів немає;

г) ${\displaystyle x^2+4x+1=}$${\displaystyle 0}$;

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

${\displaystyle D=}$${\displaystyle b^2-4ac=}$${\displaystyle 4^2-4\cdot 1\cdot 1=}$${\displaystyle 16-4=}$${\displaystyle 12}$;

${\displaystyle D>0}$, два корені:

${\displaystyle x_{1,2}=}$${\displaystyle \frac{-b\pm \sqrt{D}}{2a}=}$${\displaystyle \frac{-4\pm \sqrt{12}}{2}=}$${\displaystyle \frac{-4\pm 2\sqrt{3}}{2}=}$${\displaystyle -2\pm \sqrt{3}}$;

ґ) ${\displaystyle -x^2+2x+5=}$${\displaystyle 0\mid \cdot \left(-1\right)}$;

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

${\displaystyle a=}$${\displaystyle 1,b=}$${\displaystyle -2,c=}$${\displaystyle -5}$;

${\displaystyle D=}$${\displaystyle b^2-4ac=}$${\displaystyle {(-2)}^2-4\cdot 1\cdot (-5)=}$${\displaystyle 4+20=}$${\displaystyle 24}$;

${\displaystyle D>0}$, два корені:

${\displaystyle x_{1,2}=}$${\displaystyle \frac{-b\pm \sqrt{D}}{2a}=}$${\displaystyle \frac{2\pm \sqrt{24}}{2}=}$${\displaystyle \frac{2\pm 2\sqrt{6}}{2}=}$${\displaystyle 1\pm \sqrt{6}}$;

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

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

${\displaystyle D=}$${\displaystyle b^2-4ac=}$${\displaystyle {(-2)}^2-4\cdot 3\cdot (-3)=}$${\displaystyle 4+36=}$${\displaystyle 40}$;

${\displaystyle D>0}$, два корені:

${\displaystyle x_{1,2}=}$${\displaystyle \frac{-b\pm \sqrt{D}}{2a}=}$${\displaystyle \frac{2\pm \sqrt{40}}{6}=}$${\displaystyle \frac{2\pm 2\sqrt{10}}{6}=}$${\displaystyle \frac{1\pm \sqrt{10}}{3}}$.