999 Вправа 999Алгебра 8 клас Бевз 2025 · § 19 · стор. 187

999. Розв’яжи рівняння. а) (x² + 4x)(x² + 4x + 3) = 40; б) (x² + 3x + 1)(x² + 3x + 3) + 1 = 0; в) (x − 2)²(x² − 4x) = −3; г) (x² − 5x + 2)(x² − 5x − 1) = 28; ґ) (x + 1)(x + 2)(x + 3)(x + 4) = 120; д) (x + 3)(x + 1)(x − 5)(x − 7) = 192.

а) ${\displaystyle (x^2+4x)(x^2+4x+3)=}$${\displaystyle 40}$;

заміна ${\displaystyle x^2+4x=}$${\displaystyle t}$:

${\displaystyle t(t+3)=}$${\displaystyle 40}$;

${\displaystyle t^2+3t-40=}$${\displaystyle 0}$;

${\displaystyle a=}$${\displaystyle 1,b=}$${\displaystyle 3,c=}$${\displaystyle -40}$;

${\displaystyle D=}$${\displaystyle b^2-4ac=}$${\displaystyle 3^2-4\cdot 1\cdot (-40)=}$${\displaystyle 9+160=}$${\displaystyle 169}$;

${\displaystyle D>0}$, два корені:

${\displaystyle t_{1,2}=}$${\displaystyle \frac{-b\pm \sqrt{D}}{2a}=}$${\displaystyle \frac{-3\pm \sqrt{169}}{2}=}$${\displaystyle \frac{-3\pm 13}{2}}$;

${\displaystyle t_1=}$${\displaystyle \frac{-3+13}{2}=}$${\displaystyle 5}$;

${\displaystyle t_2=}$${\displaystyle \frac{-3-13}{2}=}$${\displaystyle -8}$;

${\displaystyle x^2+4x=}$${\displaystyle 5}$:

${\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}$;

${\displaystyle x^2+4x=}$${\displaystyle -8}$:

${\displaystyle x^2+4x+8=}$${\displaystyle 0}$;

${\displaystyle a=}$${\displaystyle 1,b=}$${\displaystyle 4,c=}$${\displaystyle 8}$;

${\displaystyle D=}$${\displaystyle b^2-4ac=}$${\displaystyle 4^2-4\cdot 1\cdot 8=}$${\displaystyle 16-32=}$${\displaystyle -16}$;

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

Корені: ${\displaystyle -5;1}$;

б) ${\displaystyle (x^2+3x+1)(x^2+3x+3)+1=}$${\displaystyle 0}$;

заміна ${\displaystyle x^2+3x=}$${\displaystyle t}$:

${\displaystyle (t+1)(t+3)+1=}$${\displaystyle 0}$;

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

${\displaystyle t^2+4t+4=}$${\displaystyle 0}$;

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

${\displaystyle D=}$${\displaystyle b^2-4ac=}$${\displaystyle 4^2-4\cdot 1\cdot 4=}$${\displaystyle 16-16=}$${\displaystyle 0}$;

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

${\displaystyle t=}$${\displaystyle -\frac{b}{2a}=}$${\displaystyle \frac{-4}{2}=}$${\displaystyle -2}$;

${\displaystyle x^2+3x=}$${\displaystyle -2}$:

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

${\displaystyle a=}$${\displaystyle 1,b=}$${\displaystyle 3,c=}$${\displaystyle 2}$;

${\displaystyle D=}$${\displaystyle b^2-4ac=}$${\displaystyle 3^2-4\cdot 1\cdot 2=}$${\displaystyle 9-8=}$${\displaystyle 1}$;

${\displaystyle D>0}$, два корені:

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

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

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

в) ${\displaystyle {(x-2)}^2(x^2-4x)=}$${\displaystyle -3}$;

${\displaystyle {(x-2)}^2=}$${\displaystyle x^2-4x+4}$;

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

заміна ${\displaystyle x^2-4x=}$${\displaystyle t}$:

${\displaystyle (t+4)t=}$${\displaystyle -3}$;

${\displaystyle t^2+4t+3=}$${\displaystyle 0}$;

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

${\displaystyle D=}$${\displaystyle b^2-4ac=}$${\displaystyle 4^2-4\cdot 1\cdot 3=}$${\displaystyle 16-12=}$${\displaystyle 4}$;

${\displaystyle D>0}$, два корені:

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

${\displaystyle t_1=}$${\displaystyle \frac{-4+2}{2}=}$${\displaystyle -1}$;

${\displaystyle t_2=}$${\displaystyle \frac{-4-2}{2}=}$${\displaystyle -3}$;

${\displaystyle x^2-4x=}$${\displaystyle -1}$:

${\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}}$;

${\displaystyle x^2-4x=}$${\displaystyle -3}$:

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

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

${\displaystyle D=}$${\displaystyle b^2-4ac=}$${\displaystyle {(-4)}^2-4\cdot 1\cdot 3=}$${\displaystyle 16-12=}$${\displaystyle 4}$;

${\displaystyle D>0}$, два корені:

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

${\displaystyle x_1=}$${\displaystyle \frac{4+2}{2}=}$${\displaystyle 3}$;

${\displaystyle x_2=}$${\displaystyle \frac{4-2}{2}=}$${\displaystyle 1}$;

Корені: ${\displaystyle 1;3;2\pm \sqrt{3}}$;

г) ${\displaystyle (x^2-5x+2)(x^2-5x-1)=}$${\displaystyle 28}$;

заміна ${\displaystyle x^2-5x=}$${\displaystyle t}$:

${\displaystyle (t+2)(t-1)=}$${\displaystyle 28}$;

${\displaystyle t^2+t-2-28=}$${\displaystyle 0}$;

${\displaystyle t^2+t-30=}$${\displaystyle 0}$;

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

${\displaystyle D=}$${\displaystyle b^2-4ac=}$${\displaystyle 1^2-4\cdot 1\cdot (-30)=}$${\displaystyle 1+120=}$${\displaystyle 121}$;

${\displaystyle D>0}$, два корені:

${\displaystyle t_{1,2}=}$${\displaystyle \frac{-b\pm \sqrt{D}}{2a}=}$${\displaystyle \frac{-1\pm \sqrt{121}}{2}=}$${\displaystyle \frac{-1\pm 11}{2}}$;

${\displaystyle t_1=}$${\displaystyle \frac{-1+11}{2}=}$${\displaystyle 5}$;

${\displaystyle t_2=}$${\displaystyle \frac{-1-11}{2}=}$${\displaystyle -6}$;

${\displaystyle x^2-5x=}$${\displaystyle 5}$:

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

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

${\displaystyle D=}$${\displaystyle b^2-4ac=}$${\displaystyle {(-5)}^2-4\cdot 1\cdot (-5)=}$${\displaystyle 25+20=}$${\displaystyle 45}$;

${\displaystyle D>0}$, два корені:

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

${\displaystyle x^2-5x=}$${\displaystyle -6}$:

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

${\displaystyle a=}$${\displaystyle 1,b=}$${\displaystyle -5,c=}$${\displaystyle 6}$;

${\displaystyle D=}$${\displaystyle b^2-4ac=}$${\displaystyle {(-5)}^2-4\cdot 1\cdot 6=}$${\displaystyle 25-24=}$${\displaystyle 1}$;

${\displaystyle D>0}$, два корені:

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

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

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

Корені: ${\displaystyle 2;3;\frac{5\pm 3\sqrt{5}}{2}}$;

ґ) ${\displaystyle (x+1)(x+2)(x+3)(x+4)=}$${\displaystyle 120}$;

групуємо: ${\displaystyle (x+1)(x+4)=}$${\displaystyle x^2+5x+4}$;

${\displaystyle (x+2)(x+3)=}$${\displaystyle x^2+5x+6}$;

${\displaystyle (x^2+5x+4)(x^2+5x+6)=}$${\displaystyle 120}$;

заміна ${\displaystyle x^2+5x=}$${\displaystyle t}$:

${\displaystyle (t+4)(t+6)=}$${\displaystyle 120}$;

${\displaystyle t^2+10t+24-120=}$${\displaystyle 0}$;

${\displaystyle t^2+10t-96=}$${\displaystyle 0}$;

${\displaystyle a=}$${\displaystyle 1,b=}$${\displaystyle 10,c=}$${\displaystyle -96}$;

${\displaystyle D=}$${\displaystyle b^2-4ac=}$${\displaystyle 10^2-4\cdot 1\cdot (-96)=}$${\displaystyle 100+384=}$${\displaystyle 484}$;

${\displaystyle D>0}$, два корені:

${\displaystyle t_{1,2}=}$${\displaystyle \frac{-b\pm \sqrt{D}}{2a}=}$${\displaystyle \frac{-10\pm \sqrt{484}}{2}=}$${\displaystyle \frac{-10\pm 22}{2}}$;

${\displaystyle t_1=}$${\displaystyle \frac{-10+22}{2}=}$${\displaystyle 6}$;

${\displaystyle t_2=}$${\displaystyle \frac{-10-22}{2}=}$${\displaystyle -16}$;

${\displaystyle x^2+5x=}$${\displaystyle 6}$:

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

${\displaystyle a=}$${\displaystyle 1,b=}$${\displaystyle 5,c=}$${\displaystyle -6}$;

${\displaystyle D=}$${\displaystyle b^2-4ac=}$${\displaystyle 5^2-4\cdot 1\cdot (-6)=}$${\displaystyle 25+24=}$${\displaystyle 49}$;

${\displaystyle D>0}$, два корені:

${\displaystyle x_{1,2}=}$${\displaystyle \frac{-b\pm \sqrt{D}}{2a}=}$${\displaystyle \frac{-5\pm \sqrt{49}}{2}=}$${\displaystyle \frac{-5\pm 7}{2}}$;

${\displaystyle x_1=}$${\displaystyle \frac{-5+7}{2}=}$${\displaystyle 1}$;

${\displaystyle x_2=}$${\displaystyle \frac{-5-7}{2}=}$${\displaystyle -6}$;

${\displaystyle x^2+5x=}$${\displaystyle -16}$:

${\displaystyle x^2+5x+16=}$${\displaystyle 0}$;

${\displaystyle a=}$${\displaystyle 1,b=}$${\displaystyle 5,c=}$${\displaystyle 16}$;

${\displaystyle D=}$${\displaystyle b^2-4ac=}$${\displaystyle 5^2-4\cdot 1\cdot 16=}$${\displaystyle 25-64=}$${\displaystyle -39}$;

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

Корені: ${\displaystyle -6;1}$;

д) ${\displaystyle (x+3)(x+1)(x-5)(x-7)=}$${\displaystyle 192}$;

групуємо: ${\displaystyle (x+3)(x-7)=}$${\displaystyle x^2-4x-21}$;

${\displaystyle (x+1)(x-5)=}$${\displaystyle x^2-4x-5}$;

${\displaystyle (x^2-4x-21)(x^2-4x-5)=}$${\displaystyle 192}$;

заміна ${\displaystyle x^2-4x=}$${\displaystyle t}$:

${\displaystyle (t-21)(t-5)=}$${\displaystyle 192}$;

${\displaystyle t^2-26t+105-192=}$${\displaystyle 0}$;

${\displaystyle t^2-26t-87=}$${\displaystyle 0}$;

${\displaystyle a=}$${\displaystyle 1,b=}$${\displaystyle -26,c=}$${\displaystyle -87}$;

${\displaystyle D=}$${\displaystyle b^2-4ac=}$${\displaystyle {(-26)}^2-4\cdot 1\cdot (-87)=}$${\displaystyle 676+348=}$${\displaystyle 1024}$;

${\displaystyle D>0}$, два корені:

${\displaystyle t_{1,2}=}$${\displaystyle \frac{-b\pm \sqrt{D}}{2a}=}$${\displaystyle \frac{26\pm \sqrt{1024}}{2}=}$${\displaystyle \frac{26\pm 32}{2}}$;

${\displaystyle t_1=}$${\displaystyle \frac{26+32}{2}=}$${\displaystyle 29}$;

${\displaystyle t_2=}$${\displaystyle \frac{26-32}{2}=}$${\displaystyle -3}$;

${\displaystyle x^2-4x=}$${\displaystyle 29}$:

${\displaystyle x^2-4x-29=}$${\displaystyle 0}$;

${\displaystyle a=}$${\displaystyle 1,b=}$${\displaystyle -4,c=}$${\displaystyle -29}$;

${\displaystyle D=}$${\displaystyle b^2-4ac=}$${\displaystyle {(-4)}^2-4\cdot 1\cdot (-29)=}$${\displaystyle 16+116=}$${\displaystyle 132}$;

${\displaystyle D>0}$, два корені:

${\displaystyle x_{1,2}=}$${\displaystyle \frac{-b\pm \sqrt{D}}{2a}=}$${\displaystyle \frac{4\pm \sqrt{132}}{2}=}$${\displaystyle \frac{4\pm 2\sqrt{33}}{2}=}$${\displaystyle 2\pm \sqrt{33}}$;

${\displaystyle x^2-4x=}$${\displaystyle -3}$:

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

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

${\displaystyle D=}$${\displaystyle b^2-4ac=}$${\displaystyle {(-4)}^2-4\cdot 1\cdot 3=}$${\displaystyle 16-12=}$${\displaystyle 4}$;

${\displaystyle D>0}$, два корені:

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

${\displaystyle x_1=}$${\displaystyle \frac{4+2}{2}=}$${\displaystyle 3}$;

${\displaystyle x_2=}$${\displaystyle \frac{4-2}{2}=}$${\displaystyle 1}$;

Корені: ${\displaystyle 1;3;2\pm \sqrt{33}}$.