758° Вправа 758Алгебра 8 клас Тарасенкова 2025 · § 18 · стор. 226

758. Розв’яжіть квадратне рівняння:

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

${\displaystyle D=}$${\displaystyle 3^2-4\cdot 2\cdot (-5)=}$${\displaystyle 9+40=}$${\displaystyle 49}$;

${\displaystyle x_{1,2}=}$${\displaystyle \frac{-3\pm \sqrt{49}}{2\cdot 2}=}$${\displaystyle \frac{-3\pm 7}{4}}$;

${\displaystyle x_1=}$${\displaystyle \frac{-3+7}{4}=}$${\displaystyle 1}$;

${\displaystyle x_2=}$${\displaystyle \frac{-3-7}{4}=}$${\displaystyle \frac{-10}{4}=}$${\displaystyle -2,5}$;

2) ${\displaystyle 9x^2-8x-1=}$${\displaystyle 0}$;

${\displaystyle D=}$${\displaystyle {(-8)}^2-4\cdot 9\cdot (-1)=}$${\displaystyle 64+36=}$${\displaystyle 100}$;

${\displaystyle x_{1,2}=}$${\displaystyle \frac{8\pm \sqrt{100}}{2\cdot 9}=}$${\displaystyle \frac{8\pm 10}{18}}$;

${\displaystyle x_1=}$${\displaystyle \frac{8+10}{18}=}$${\displaystyle 1}$;

${\displaystyle x_2=}$${\displaystyle \frac{8-10}{18}=}$${\displaystyle \frac{-2}{18}=}$${\displaystyle -\frac{1}{9}}$;

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

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

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

${\displaystyle x_1=}$${\displaystyle \frac{-1+9}{-4}=}$${\displaystyle -2}$;

${\displaystyle x_2=}$${\displaystyle \frac{-1-9}{-4}=}$${\displaystyle \frac{-10}{-4}=}$${\displaystyle 2,5}$;

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

${\displaystyle D=}$${\displaystyle 5^2-4\cdot 3\cdot (-2)=}$${\displaystyle 25+24=}$${\displaystyle 49}$;

${\displaystyle x_{1,2}=}$${\displaystyle \frac{-5\pm \sqrt{49}}{2\cdot 3}=}$${\displaystyle \frac{-5\pm 7}{6}}$;

${\displaystyle x_1=}$${\displaystyle \frac{-5+7}{6}=}$${\displaystyle \frac{2}{6}=}$${\displaystyle \frac{1}{3}}$;

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

5) ${\displaystyle 5x^2+6x+1=}$${\displaystyle 0}$;

${\displaystyle D=}$${\displaystyle 6^2-4\cdot 5\cdot 1=}$${\displaystyle 36-20=}$${\displaystyle 16}$;

${\displaystyle x_{1,2}=}$${\displaystyle \frac{-6\pm \sqrt{16}}{2\cdot 5}=}$${\displaystyle \frac{-6\pm 4}{10}}$;

${\displaystyle x_1=}$${\displaystyle \frac{-6+4}{10}=}$${\displaystyle \frac{-2}{10}=}$${\displaystyle -0,2}$;

${\displaystyle x_2=}$${\displaystyle \frac{-6-4}{10}=}$${\displaystyle -1}$;

6) ${\displaystyle -4x^2+28x-49=}$${\displaystyle 0}$;

${\displaystyle D=}$${\displaystyle {28}^2-4\cdot (-4)\cdot (-49)=}$${\displaystyle 784-784=}$${\displaystyle 0}$;

${\displaystyle x=}$${\displaystyle \frac{-28}{-8}=}$${\displaystyle 3,5}$;

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

${\displaystyle D=}$${\displaystyle {(-5)}^2-4\cdot 2\cdot (-3)=}$${\displaystyle 25+24=}$${\displaystyle 49}$;

${\displaystyle x_{1,2}=}$${\displaystyle \frac{5\pm \sqrt{49}}{2\cdot 2}=}$${\displaystyle \frac{5\pm 7}{4}}$;

${\displaystyle x_1=}$${\displaystyle \frac{5+7}{4}=}$${\displaystyle 3}$;

${\displaystyle x_2=}$${\displaystyle \frac{5-7}{4}=}$${\displaystyle \frac{-2}{4}=}$${\displaystyle -0,5}$;

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

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

${\displaystyle x_{1,2}=}$${\displaystyle \frac{-3\pm \sqrt{1}}{2\cdot (-2)}=}$${\displaystyle \frac{-3\pm 1}{-4}}$;

${\displaystyle x_1=}$${\displaystyle \frac{-3+1}{-4}=}$${\displaystyle \frac{-2}{-4}=}$${\displaystyle 0,5}$;

${\displaystyle x_2=}$${\displaystyle \frac{-3-1}{-4}=}$${\displaystyle 1}$.