1076 Вправа 1076Алгебра 8 клас Бевз 2025 · § 21 · стор. 203

1076. Знайди корені квадратного тричлена. а) 4x² + 3x − 1; б) 6x² + 7x − 5; в) −x² − 4x + 5; г) 9x² + 6x + 1; ґ) −4x² + 5x − 2; д) 0,4x² + 0,7x − 3.

а) ${\displaystyle 4x^2+3x-1}$:

${\displaystyle 4x^2+3x-1=}$${\displaystyle 0}$;

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

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

${\displaystyle D>0}$, два корені:

${\displaystyle x_{1,2}=}$${\displaystyle \frac{-b\pm \sqrt{D}}{2a}=}$${\displaystyle \frac{-3\pm \sqrt{25}}{8}=}$${\displaystyle \frac{-3\pm 5}{8}}$;

${\displaystyle x_1=}$${\displaystyle \frac{-3+5}{8}=}$${\displaystyle 0,25}$;

${\displaystyle x_2=}$${\displaystyle \frac{-3-5}{8}=}$${\displaystyle -1}$;

б) ${\displaystyle 6x^2+7x-5}$:

${\displaystyle 6x^2+7x-5=}$${\displaystyle 0}$;

${\displaystyle a=}$${\displaystyle 6,b=}$${\displaystyle 7,c=}$${\displaystyle -5}$;

${\displaystyle D=}$${\displaystyle b^2-4ac=}$${\displaystyle 7^2-4\cdot 6\cdot (-5)=}$${\displaystyle 49+120=}$${\displaystyle 169}$;

${\displaystyle D>0}$, два корені:

${\displaystyle x_{1,2}=}$${\displaystyle \frac{-b\pm \sqrt{D}}{2a}=}$${\displaystyle \frac{-7\pm \sqrt{169}}{12}=}$${\displaystyle \frac{-7\pm 13}{12}}$;

${\displaystyle x_1=}$${\displaystyle \frac{-7+13}{12}=}$${\displaystyle 0,5}$;

${\displaystyle x_2=}$${\displaystyle \frac{-7-13}{12}=}$${\displaystyle -\frac{5}{3}}$;

в) ${\displaystyle -x^2-4x+5}$:

${\displaystyle -x^2-4x+5=}$${\displaystyle 0\mid \cdot \left(-1\right)}$;

${\displaystyle x^2+4x-5=}$${\displaystyle 0}$;

${\displaystyle a=}$${\displaystyle 1,b=}$${\displaystyle 4,c=}$${\displaystyle -5}$;

${\displaystyle D=}$${\displaystyle b^2-4ac=}$${\displaystyle 4^2-4\cdot 1\cdot (-5)=}$${\displaystyle 16+20=}$${\displaystyle 36}$;

${\displaystyle D>0}$, два корені:

${\displaystyle x_{1,2}=}$${\displaystyle \frac{-b\pm \sqrt{D}}{2a}=}$${\displaystyle \frac{-4\pm \sqrt{36}}{2}=}$${\displaystyle \frac{-4\pm 6}{2}}$;

${\displaystyle x_1=}$${\displaystyle \frac{-4+6}{2}=}$${\displaystyle 1}$;

${\displaystyle x_2=}$${\displaystyle \frac{-4-6}{2}=}$${\displaystyle -5}$;

г) ${\displaystyle 9x^2+6x+1}$:

${\displaystyle 9x^2+6x+1=}$${\displaystyle 0}$;

${\displaystyle a=}$${\displaystyle 9,b=}$${\displaystyle 6,c=}$${\displaystyle 1}$;

${\displaystyle D=}$${\displaystyle b^2-4ac=}$${\displaystyle 6^2-4\cdot 9\cdot 1=}$${\displaystyle 36-36=}$${\displaystyle 0}$;

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

${\displaystyle x=}$${\displaystyle -\frac{b}{2a}=}$${\displaystyle \frac{-6}{18}=}$${\displaystyle -\frac{1}{3}}$;

ґ) ${\displaystyle -4x^2+5x-2}$:

${\displaystyle -4x^2+5x-2=}$${\displaystyle 0\mid \cdot \left(-1\right)}$;

${\displaystyle 4x^2-5x+2=}$${\displaystyle 0}$;

${\displaystyle a=}$${\displaystyle 4,b=}$${\displaystyle -5,c=}$${\displaystyle 2}$;

${\displaystyle D=}$${\displaystyle b^2-4ac=}$${\displaystyle {(-5)}^2-4\cdot 4\cdot 2=}$${\displaystyle 25-32=}$${\displaystyle -7}$;

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

д) ${\displaystyle 0,4x^2+0,7x-3}$:

${\displaystyle 0,4x^2+0,7x-3=}$${\displaystyle 0\mid \cdot 10}$;

${\displaystyle 4x^2+7x-30=}$${\displaystyle 0}$;

${\displaystyle a=}$${\displaystyle 4,b=}$${\displaystyle 7,c=}$${\displaystyle -30}$;

${\displaystyle D=}$${\displaystyle b^2-4ac=}$${\displaystyle 7^2-4\cdot 4\cdot (-30)=}$${\displaystyle 49+480=}$${\displaystyle 529}$;

${\displaystyle D>0}$, два корені:

${\displaystyle x_{1,2}=}$${\displaystyle \frac{-b\pm \sqrt{D}}{2a}=}$${\displaystyle \frac{-7\pm \sqrt{529}}{8}=}$${\displaystyle \frac{-7\pm 23}{8}}$;

${\displaystyle x_1=}$${\displaystyle \frac{-7+23}{8}=}$${\displaystyle 2}$;

${\displaystyle x_2=}$${\displaystyle \frac{-7-23}{8}=}$${\displaystyle -3,75}$.