757° Вправа 757Алгебра 8 клас Тарасенкова 2025 · § 18 · стор. 226

757. Розв’яжіть квадратне рівняння:

1) ${\displaystyle 2x^2-x-6=}$${\displaystyle 0}$;

${\displaystyle D=}$${\displaystyle {(-1)}^2-4\cdot 2\cdot (-6)=}$${\displaystyle 1+48=}$${\displaystyle 49}$;

${\displaystyle x_{1,2}=}$${\displaystyle \frac{1\pm \sqrt{49}}{2\cdot 2}=}$${\displaystyle \frac{1\pm 7}{4}}$;

${\displaystyle x_1=}$${\displaystyle \frac{1+7}{4}=}$${\displaystyle 2}$;

${\displaystyle x_2=}$${\displaystyle \frac{1-7}{4}=}$${\displaystyle \frac{-6}{4}=}$${\displaystyle -1,5}$;

2) ${\displaystyle 9x^2-6x-8=}$${\displaystyle 0}$;

${\displaystyle D=}$${\displaystyle {(-6)}^2-4\cdot 9\cdot (-8)=}$${\displaystyle 36+288=}$${\displaystyle 324}$;

${\displaystyle x_{1,2}=}$${\displaystyle \frac{6\pm \sqrt{324}}{2\cdot 9}=}$${\displaystyle \frac{6\pm 18}{18}}$;

${\displaystyle x_1=}$${\displaystyle \frac{6+18}{18}=}$${\displaystyle \frac{24}{18}=}$${\displaystyle 1\frac{1}{3}}$;

${\displaystyle x_2=}$${\displaystyle \frac{6-18}{18}=}$${\displaystyle \frac{-12}{18}=}$${\displaystyle -\frac{2}{3}}$;

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

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

${\displaystyle x_{1,2}=}$${\displaystyle \frac{-7\pm \sqrt{169}}{2\cdot 5}=}$${\displaystyle \frac{-7\pm 13}{10}}$;

${\displaystyle x_1=}$${\displaystyle \frac{-7+13}{10}=}$${\displaystyle \frac{6}{10}=}$${\displaystyle 0,6}$;

${\displaystyle x_2=}$${\displaystyle \frac{-7-13}{10}=}$${\displaystyle -2}$;

4) ${\displaystyle 4x^2-8x+3=}$${\displaystyle 0}$;

${\displaystyle D=}$${\displaystyle {(-8)}^2-4\cdot 4\cdot 3=}$${\displaystyle 64-48=}$${\displaystyle 16}$;

${\displaystyle x_{1,2}=}$${\displaystyle \frac{8\pm \sqrt{16}}{2\cdot 4}=}$${\displaystyle \frac{8\pm 4}{8}}$;

${\displaystyle x_1=}$${\displaystyle \frac{8+4}{8}=}$${\displaystyle \frac{12}{8}=}$${\displaystyle 1,5}$;

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

5) ${\displaystyle -4x^2-11x+3=}$${\displaystyle 0}$;

${\displaystyle D=}$${\displaystyle {(-11)}^2-4\cdot (-4)\cdot 3=}$${\displaystyle 121+48=}$${\displaystyle 169}$;

${\displaystyle x_{1,2}=}$${\displaystyle \frac{11\pm \sqrt{169}}{2\cdot (-4)}=}$${\displaystyle \frac{11\pm 13}{-8}}$;

${\displaystyle x_1=}$${\displaystyle \frac{11+13}{-8}=}$${\displaystyle -3}$;

${\displaystyle x_2=}$${\displaystyle \frac{11-13}{-8}=}$${\displaystyle \frac{-2}{-8}=}$${\displaystyle 0,25}$;

6) ${\displaystyle 5x^2+14x-3=}$${\displaystyle 0}$;

${\displaystyle D=}$${\displaystyle {14}^2-4\cdot 5\cdot (-3)=}$${\displaystyle 196+60=}$${\displaystyle 256}$;

${\displaystyle x_{1,2}=}$${\displaystyle \frac{-14\pm \sqrt{256}}{2\cdot 5}=}$${\displaystyle \frac{-14\pm 16}{10}}$;

${\displaystyle x_1=}$${\displaystyle \frac{-14+16}{10}=}$${\displaystyle \frac{2}{10}=}$${\displaystyle 0,2}$;

${\displaystyle x_2=}$${\displaystyle \frac{-14-16}{10}=}$${\displaystyle -3}$;

7) ${\displaystyle -6x^2-x+1=}$${\displaystyle 0}$;

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

${\displaystyle x_{1,2}=}$${\displaystyle \frac{1\pm \sqrt{25}}{2\cdot (-6)}=}$${\displaystyle \frac{1\pm 5}{-12}}$;

${\displaystyle x_1=}$${\displaystyle \frac{1+5}{-12}=}$${\displaystyle \frac{6}{-12}=}$${\displaystyle -0,5}$;

${\displaystyle x_2=}$${\displaystyle \frac{1-5}{-12}=}$${\displaystyle \frac{-4}{-12}=}$${\displaystyle \frac{1}{3}}$;

8) ${\displaystyle 2x^2-5x+3=}$${\displaystyle 0}$;

${\displaystyle D=}$${\displaystyle {(-5)}^2-4\cdot 2\cdot 3=}$${\displaystyle 25-24=}$${\displaystyle 1}$;

${\displaystyle x_{1,2}=}$${\displaystyle \frac{5\pm \sqrt{1}}{2\cdot 2}=}$${\displaystyle \frac{5\pm 1}{4}}$;

${\displaystyle x_1=}$${\displaystyle \frac{5+1}{4}=}$${\displaystyle \frac{6}{4}=}$${\displaystyle 1,5}$;

${\displaystyle x_2=}$${\displaystyle \frac{5-1}{4}=}$${\displaystyle 1}$;

9) ${\displaystyle 2x^2-9x+10=}$${\displaystyle 0}$;

${\displaystyle D=}$${\displaystyle {(-9)}^2-4\cdot 2\cdot 10=}$${\displaystyle 81-80=}$${\displaystyle 1}$;

${\displaystyle x_{1,2}=}$${\displaystyle \frac{9\pm \sqrt{1}}{2\cdot 2}=}$${\displaystyle \frac{9\pm 1}{4}}$;

${\displaystyle x_1=}$${\displaystyle \frac{9+1}{4}=}$${\displaystyle \frac{10}{4}=}$${\displaystyle 2,5}$;

${\displaystyle x_2=}$${\displaystyle \frac{9-1}{4}=}$${\displaystyle 2}$;

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

${\displaystyle D=}$${\displaystyle {10}^2-4\cdot (-25)\cdot 3=}$${\displaystyle 100+300=}$${\displaystyle 400}$;

${\displaystyle x_{1,2}=}$${\displaystyle \frac{-10\pm \sqrt{400}}{2\cdot (-25)}=}$${\displaystyle \frac{-10\pm 20}{-50}}$;

${\displaystyle x_1=}$${\displaystyle \frac{-10+20}{-50}=}$${\displaystyle \frac{10}{-50}=}$${\displaystyle -0,2}$;

${\displaystyle x_2=}$${\displaystyle \frac{-10-20}{-50}=}$${\displaystyle \frac{-30}{-50}=}$${\displaystyle 0,6}$;

11) ${\displaystyle 16x^2+56x+45=}$${\displaystyle 0}$;

${\displaystyle D=}$${\displaystyle {56}^2-4\cdot 16\cdot 45=}$${\displaystyle 3136-2880=}$${\displaystyle 256}$;

${\displaystyle x_{1,2}=}$${\displaystyle \frac{-56\pm \sqrt{256}}{2\cdot 16}=}$${\displaystyle \frac{-56\pm 16}{32}}$;

${\displaystyle x_1=}$${\displaystyle \frac{-56+16}{32}=}$${\displaystyle \frac{-40}{32}=}$${\displaystyle -1,25}$;

${\displaystyle x_2=}$${\displaystyle \frac{-56-16}{32}=}$${\displaystyle \frac{-72}{32}=}$${\displaystyle -2,25}$;

12) ${\displaystyle -4x^2+7x-3=}$${\displaystyle 0}$;

${\displaystyle D=}$${\displaystyle 7^2-4\cdot (-4)\cdot (-3)=}$${\displaystyle 49-48=}$${\displaystyle 1}$;

${\displaystyle x_{1,2}=}$${\displaystyle \frac{-7\pm \sqrt{1}}{2\cdot (-4)}=}$${\displaystyle \frac{-7\pm 1}{-8}}$;

${\displaystyle x_1=}$${\displaystyle \frac{-7+1}{-8}=}$${\displaystyle \frac{-6}{-8}=}$${\displaystyle 0,75}$;

${\displaystyle x_2=}$${\displaystyle \frac{-7-1}{-8}=}$${\displaystyle 1}$;

13) ${\displaystyle 9x^2+48x+64=}$${\displaystyle 0}$;

${\displaystyle D=}$${\displaystyle {48}^2-4\cdot 9\cdot 64=}$${\displaystyle 2304-2304=}$${\displaystyle 0}$;

${\displaystyle x=}$${\displaystyle \frac{-48}{18}=}$${\displaystyle -2\frac{2}{3}}$;

14) ${\displaystyle -3x^2+19x-6=}$${\displaystyle 0}$;

${\displaystyle D=}$${\displaystyle {19}^2-4\cdot (-3)\cdot (-6)=}$${\displaystyle 361-72=}$${\displaystyle 289}$;

${\displaystyle x_{1,2}=}$${\displaystyle \frac{-19\pm \sqrt{289}}{2\cdot (-3)}=}$${\displaystyle \frac{-19\pm 17}{-6}}$;

${\displaystyle x_1=}$${\displaystyle \frac{-19+17}{-6}=}$${\displaystyle \frac{-2}{-6}=}$${\displaystyle \frac{1}{3}}$;

${\displaystyle x_2=}$${\displaystyle \frac{-19-17}{-6}=}$${\displaystyle 6}$;

15) ${\displaystyle 2x^2-11x+5=}$${\displaystyle 0}$;

${\displaystyle D=}$${\displaystyle {(-11)}^2-4\cdot 2\cdot 5=}$${\displaystyle 121-40=}$${\displaystyle 81}$;

${\displaystyle x_{1,2}=}$${\displaystyle \frac{11\pm \sqrt{81}}{2\cdot 2}=}$${\displaystyle \frac{11\pm 9}{4}}$;

${\displaystyle x_1=}$${\displaystyle \frac{11+9}{4}=}$${\displaystyle 5}$;

${\displaystyle x_2=}$${\displaystyle \frac{11-9}{4}=}$${\displaystyle \frac{2}{4}=}$${\displaystyle 0,5}$;

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

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

${\displaystyle x=}$${\displaystyle \frac{-4}{-8}=}$${\displaystyle 0,5}$.