965 Вправа 965Алгебра 8 клас Бевз 2025 · § 19 · стор. 183

965. Розв’яжи рівняння, що зводиться до квадратного. а) 4x(x − 1) = 3; б) 3x(3x − 8) = 20; в) 3x(2x − 5) = 2(x² + 2); г) 2x(3x + 4) = 4x² + 5x + 27; ґ) (x + 4)(2x − 3) − (5x − 6)(x − 3) = 10.

Розкриваємо дужки, переносимо все в ліву частину і зводимо подібні.

а) ${\displaystyle 4x(x-1)=}$${\displaystyle 3}$;

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

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

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

${\displaystyle D=}$${\displaystyle b^2-4ac=}$${\displaystyle {(-4)}^2-4\cdot 4\cdot (-3)=}$${\displaystyle 16+48=}$${\displaystyle 64}$;

${\displaystyle D>0}$, два корені:

${\displaystyle x_{1,2}=}$${\displaystyle \frac{-b\pm \sqrt{D}}{2a}=}$${\displaystyle \frac{4\pm \sqrt{64}}{8}=}$${\displaystyle \frac{4\pm 8}{8}}$;

${\displaystyle x_1=}$${\displaystyle \frac{4+8}{8}=}$${\displaystyle 1,5}$;

${\displaystyle x_2=}$${\displaystyle \frac{4-8}{8}=}$${\displaystyle -0,5}$;

б) ${\displaystyle 3x(3x-8)=}$${\displaystyle 20}$;

${\displaystyle 9x^2-24x=}$${\displaystyle 20}$;

${\displaystyle 9x^2-24x-20=}$${\displaystyle 0}$;

${\displaystyle a=}$${\displaystyle 9,b=}$${\displaystyle -24,c=}$${\displaystyle -20}$;

${\displaystyle D=}$${\displaystyle b^2-4ac=}$${\displaystyle {(-24)}^2-4\cdot 9\cdot (-20)=}$${\displaystyle 576+720=}$${\displaystyle 1296}$;

${\displaystyle D>0}$, два корені:

${\displaystyle x_{1,2}=}$${\displaystyle \frac{-b\pm \sqrt{D}}{2a}=}$${\displaystyle \frac{24\pm \sqrt{1296}}{18}=}$${\displaystyle \frac{24\pm 36}{18}}$;

${\displaystyle x_1=}$${\displaystyle \frac{24+36}{18}=}$${\displaystyle \frac{10}{3}}$;

${\displaystyle x_2=}$${\displaystyle \frac{24-36}{18}=}$${\displaystyle -\frac{2}{3}}$;

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

${\displaystyle 6x^2-15x=}$${\displaystyle 2x^2+4}$;

${\displaystyle 6x^2-15x-2x^2-4=}$${\displaystyle 0}$;

${\displaystyle 4x^2-15x-4=}$${\displaystyle 0}$;

${\displaystyle a=}$${\displaystyle 4,b=}$${\displaystyle -15,c=}$${\displaystyle -4}$;

${\displaystyle D=}$${\displaystyle b^2-4ac=}$${\displaystyle {(-15)}^2-4\cdot 4\cdot (-4)=}$${\displaystyle 225+64=}$${\displaystyle 289}$;

${\displaystyle D>0}$, два корені:

${\displaystyle x_{1,2}=}$${\displaystyle \frac{-b\pm \sqrt{D}}{2a}=}$${\displaystyle \frac{15\pm \sqrt{289}}{8}=}$${\displaystyle \frac{15\pm 17}{8}}$;

${\displaystyle x_1=}$${\displaystyle \frac{15+17}{8}=}$${\displaystyle 4}$;

${\displaystyle x_2=}$${\displaystyle \frac{15-17}{8}=}$${\displaystyle -0,25}$;

г) ${\displaystyle 2x(3x+4)=}$${\displaystyle 4x^2+5x+27}$;

${\displaystyle 6x^2+8x=}$${\displaystyle 4x^2+5x+27}$;

${\displaystyle 6x^2+8x-4x^2-5x-27=}$${\displaystyle 0}$;

${\displaystyle 2x^2+3x-27=}$${\displaystyle 0}$;

${\displaystyle a=}$${\displaystyle 2,b=}$${\displaystyle 3,c=}$${\displaystyle -27}$;

${\displaystyle D=}$${\displaystyle b^2-4ac=}$${\displaystyle 3^2-4\cdot 2\cdot (-27)=}$${\displaystyle 9+216=}$${\displaystyle 225}$;

${\displaystyle D>0}$, два корені:

${\displaystyle x_{1,2}=}$${\displaystyle \frac{-b\pm \sqrt{D}}{2a}=}$${\displaystyle \frac{-3\pm \sqrt{225}}{4}=}$${\displaystyle \frac{-3\pm 15}{4}}$;

${\displaystyle x_1=}$${\displaystyle \frac{-3+15}{4}=}$${\displaystyle 3}$;

${\displaystyle x_2=}$${\displaystyle \frac{-3-15}{4}=}$${\displaystyle -4,5}$;

ґ) ${\displaystyle (x+4)(2x-3)-(5x-6)(x-3)=}$${\displaystyle 10}$;

${\displaystyle (2x^2-3x+8x-12)-(5x^2-15x-6x+18)=}$${\displaystyle 10}$;

${\displaystyle 2x^2+5x-12-5x^2+21x-18=}$${\displaystyle 10}$;

${\displaystyle -3x^2+26x-40=}$${\displaystyle 0}$;

${\displaystyle 3x^2-26x+40=}$${\displaystyle 0}$;

${\displaystyle a=}$${\displaystyle 3,b=}$${\displaystyle -26,c=}$${\displaystyle 40}$;

${\displaystyle D=}$${\displaystyle b^2-4ac=}$${\displaystyle {(-26)}^2-4\cdot 3\cdot 40=}$${\displaystyle 676-480=}$${\displaystyle 196}$;

${\displaystyle D>0}$, два корені:

${\displaystyle x_{1,2}=}$${\displaystyle \frac{-b\pm \sqrt{D}}{2a}=}$${\displaystyle \frac{26\pm \sqrt{196}}{6}=}$${\displaystyle \frac{26\pm 14}{6}}$;

${\displaystyle x_1=}$${\displaystyle \frac{26+14}{6}=}$${\displaystyle \frac{20}{3}}$;

${\displaystyle x_2=}$${\displaystyle \frac{26-14}{6}=}$${\displaystyle 2}$.