968 Вправа 968Алгебра 8 клас Бевз 2025 · § 19 · стор. 184

968. Розв’яжи рівняння, що зводиться до квадратного. а) (x − 5)² = 3x − 5; б) (x + 4)² = 3x² − 8; в) (p − 3)² = 2(p + 1); г) (3c − 5)² = 10c − 19.

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

${\displaystyle x^2-10x+25=}$${\displaystyle 3x-5}$;

${\displaystyle x^2-10x+25-3x+5=}$${\displaystyle 0}$;

${\displaystyle x^2-13x+30=}$${\displaystyle 0}$;

${\displaystyle a=}$${\displaystyle 1,b=}$${\displaystyle -13,c=}$${\displaystyle 30}$;

${\displaystyle D=}$${\displaystyle b^2-4ac=}$${\displaystyle {(-13)}^2-4\cdot 1\cdot 30=}$${\displaystyle 169-120=}$${\displaystyle 49}$;

${\displaystyle D>0}$, два корені:

${\displaystyle x_{1,2}=}$${\displaystyle \frac{-b\pm \sqrt{D}}{2a}=}$${\displaystyle \frac{13\pm \sqrt{49}}{2}=}$${\displaystyle \frac{13\pm 7}{2}}$;

${\displaystyle x_1=}$${\displaystyle \frac{13+7}{2}=}$${\displaystyle 10}$;

${\displaystyle x_2=}$${\displaystyle \frac{13-7}{2}=}$${\displaystyle 3}$;

б) ${\displaystyle {(x+4)}^2=}$${\displaystyle 3x^2-8}$;

${\displaystyle x^2+8x+16=}$${\displaystyle 3x^2-8}$;

${\displaystyle x^2+8x+16-3x^2+8=}$${\displaystyle 0}$;

${\displaystyle -2x^2+8x+24=}$${\displaystyle 0\mid :\left(-2\right)}$;

${\displaystyle x^2-4x-12=}$${\displaystyle 0}$;

${\displaystyle a=}$${\displaystyle 1,b=}$${\displaystyle -4,c=}$${\displaystyle -12}$;

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

${\displaystyle D>0}$, два корені:

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

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

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

в) ${\displaystyle {(p-3)}^2=}$${\displaystyle 2(p+1)}$;

${\displaystyle p^2-6p+9=}$${\displaystyle 2p+2}$;

${\displaystyle p^2-6p+9-2p-2=}$${\displaystyle 0}$;

${\displaystyle p^2-8p+7=}$${\displaystyle 0}$;

${\displaystyle a=}$${\displaystyle 1,b=}$${\displaystyle -8,c=}$${\displaystyle 7}$;

${\displaystyle D=}$${\displaystyle b^2-4ac=}$${\displaystyle {(-8)}^2-4\cdot 1\cdot 7=}$${\displaystyle 64-28=}$${\displaystyle 36}$;

${\displaystyle D>0}$, два корені:

${\displaystyle p_{1,2}=}$${\displaystyle \frac{-b\pm \sqrt{D}}{2a}=}$${\displaystyle \frac{8\pm \sqrt{36}}{2}=}$${\displaystyle \frac{8\pm 6}{2}}$;

${\displaystyle p_1=}$${\displaystyle \frac{8+6}{2}=}$${\displaystyle 7}$;

${\displaystyle p_2=}$${\displaystyle \frac{8-6}{2}=}$${\displaystyle 1}$;

г) ${\displaystyle {(3c-5)}^2=}$${\displaystyle 10c-19}$;

${\displaystyle 9c^2-30c+25=}$${\displaystyle 10c-19}$;

${\displaystyle 9c^2-30c+25-10c+19=}$${\displaystyle 0}$;

${\displaystyle 9c^2-40c+44=}$${\displaystyle 0}$;

${\displaystyle a=}$${\displaystyle 9,b=}$${\displaystyle -40,c=}$${\displaystyle 44}$;

${\displaystyle D=}$${\displaystyle b^2-4ac=}$${\displaystyle {(-40)}^2-4\cdot 9\cdot 44=}$${\displaystyle 1600-1584=}$${\displaystyle 16}$;

${\displaystyle D>0}$, два корені:

${\displaystyle c_{1,2}=}$${\displaystyle \frac{-b\pm \sqrt{D}}{2a}=}$${\displaystyle \frac{40\pm \sqrt{16}}{18}=}$${\displaystyle \frac{40\pm 4}{18}}$;

${\displaystyle c_1=}$${\displaystyle \frac{40+4}{18}=}$${\displaystyle \frac{22}{9}}$;

${\displaystyle c_2=}$${\displaystyle \frac{40-4}{18}=}$${\displaystyle 2}$.