960 Вправа 960Алгебра 8 клас Бевз 2025 · § 19 · стор. 183

960. Користуючись формулою коренів, розв’яжи рівняння. а) −x² + 8x − 12 = 0; б) 4x² − 12x + 9 = 0; в) 3x² + 2x + 2 = 0; г) x² + 2x − 1 = 0; ґ) −x² + 4x + 2 = 0; д) 9x² − 3x − 1 = 0.

а) ${\displaystyle -x^2+8x-12=}$${\displaystyle 0\mid \cdot \left(-1\right)}$;

${\displaystyle x^2-8x+12=}$${\displaystyle 0}$;

${\displaystyle a=}$${\displaystyle 1,b=}$${\displaystyle -8,c=}$${\displaystyle 12}$;

${\displaystyle D=}$${\displaystyle b^2-4ac=}$${\displaystyle {(-8)}^2-4\cdot 1\cdot 12=}$${\displaystyle 64-48=}$${\displaystyle 16}$;

${\displaystyle D>0}$, два корені:

${\displaystyle x_{1,2}=}$${\displaystyle \frac{-b\pm \sqrt{D}}{2a}=}$${\displaystyle \frac{8\pm \sqrt{16}}{2}=}$${\displaystyle \frac{8\pm 4}{2}}$;

${\displaystyle x_1=}$${\displaystyle \frac{8+4}{2}=}$${\displaystyle 6}$;

${\displaystyle x_2=}$${\displaystyle \frac{8-4}{2}=}$${\displaystyle 2}$;

б) ${\displaystyle 4x^2-12x+9=}$${\displaystyle 0}$;

${\displaystyle a=}$${\displaystyle 4,b=}$${\displaystyle -12,c=}$${\displaystyle 9}$;

${\displaystyle D=}$${\displaystyle b^2-4ac=}$${\displaystyle {(-12)}^2-4\cdot 4\cdot 9=}$${\displaystyle 144-144=}$${\displaystyle 0}$;

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

${\displaystyle x=}$${\displaystyle -\frac{b}{2a}=}$${\displaystyle \frac{12}{8}=}$${\displaystyle 1,5}$;

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

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

${\displaystyle D=}$${\displaystyle b^2-4ac=}$${\displaystyle 2^2-4\cdot 3\cdot 2=}$${\displaystyle 4-24=}$${\displaystyle -20}$;

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

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

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

${\displaystyle D=}$${\displaystyle b^2-4ac=}$${\displaystyle 2^2-4\cdot 1\cdot (-1)=}$${\displaystyle 4+4=}$${\displaystyle 8}$;

${\displaystyle D>0}$, два корені:

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

ґ) ${\displaystyle -x^2+4x+2=}$${\displaystyle 0\mid \cdot \left(-1\right)}$;

${\displaystyle x^2-4x-2=}$${\displaystyle 0}$;

${\displaystyle a=}$${\displaystyle 1,b=}$${\displaystyle -4,c=}$${\displaystyle -2}$;

${\displaystyle D=}$${\displaystyle b^2-4ac=}$${\displaystyle {(-4)}^2-4\cdot 1\cdot (-2)=}$${\displaystyle 16+8=}$${\displaystyle 24}$;

${\displaystyle D>0}$, два корені:

${\displaystyle x_{1,2}=}$${\displaystyle \frac{-b\pm \sqrt{D}}{2a}=}$${\displaystyle \frac{4\pm \sqrt{24}}{2}=}$${\displaystyle \frac{4\pm 2\sqrt{6}}{2}=}$${\displaystyle 2\pm \sqrt{6}}$;

д) ${\displaystyle 9x^2-3x-1=}$${\displaystyle 0}$;

${\displaystyle a=}$${\displaystyle 9,b=}$${\displaystyle -3,c=}$${\displaystyle -1}$;

${\displaystyle D=}$${\displaystyle b^2-4ac=}$${\displaystyle {(-3)}^2-4\cdot 9\cdot (-1)=}$${\displaystyle 9+36=}$${\displaystyle 45}$;

${\displaystyle D>0}$, два корені:

${\displaystyle x_{1,2}=}$${\displaystyle \frac{-b\pm \sqrt{D}}{2a}=}$${\displaystyle \frac{3\pm \sqrt{45}}{18}=}$${\displaystyle \frac{3\pm 3\sqrt{5}}{18}=}$${\displaystyle \frac{1\pm \sqrt{5}}{6}}$.