759 Вправа 759Алгебра 8 клас Тарасенкова 2025 · § 18 · стор. 226

759. Розв’яжіть рівняння:

1) ${\displaystyle 7x=}$${\displaystyle 3x^2+4}$;

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

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

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

${\displaystyle x_1=}$${\displaystyle \frac{7+1}{6}=}$${\displaystyle \frac{8}{6}=}$${\displaystyle 1\frac{1}{3}}$;

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

2) ${\displaystyle 6y-1=}$${\displaystyle 5y^2}$;

${\displaystyle 5y^2-6y+1=}$${\displaystyle 0}$;

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

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

${\displaystyle y_1=}$${\displaystyle \frac{6+4}{10}=}$${\displaystyle 1}$;

${\displaystyle y_2=}$${\displaystyle \frac{6-4}{10}=}$${\displaystyle \frac{2}{10}=}$${\displaystyle 0,2}$;

3) ${\displaystyle z^2-90=}$${\displaystyle z}$;

${\displaystyle z^2-z-90=}$${\displaystyle 0}$;

${\displaystyle D=}$${\displaystyle {(-1)}^2-4\cdot 1\cdot (-90)=}$${\displaystyle 1+360=}$${\displaystyle 361}$;

${\displaystyle z_{1,2}=}$${\displaystyle \frac{1\pm \sqrt{361}}{2\cdot 1}=}$${\displaystyle \frac{1\pm 19}{2}}$;

${\displaystyle z_1=}$${\displaystyle \frac{1+19}{2}=}$${\displaystyle 10}$;

${\displaystyle z_2=}$${\displaystyle \frac{1-19}{2}=}$${\displaystyle -9}$;

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

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

${\displaystyle D=}$${\displaystyle {(-8)}^2-4\cdot 5\cdot 3=}$${\displaystyle 64-60=}$${\displaystyle 4}$;

${\displaystyle x_{1,2}=}$${\displaystyle \frac{8\pm \sqrt{4}}{2\cdot 5}=}$${\displaystyle \frac{8\pm 2}{10}}$;

${\displaystyle x_1=}$${\displaystyle \frac{8+2}{10}=}$${\displaystyle 1}$;

${\displaystyle x_2=}$${\displaystyle \frac{8-2}{10}=}$${\displaystyle \frac{6}{10}=}$${\displaystyle 0,6}$;

5) ${\displaystyle 5=}$${\displaystyle 6y-y^2}$;

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

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

${\displaystyle y_{1,2}=}$${\displaystyle \frac{6\pm \sqrt{16}}{2\cdot 1}=}$${\displaystyle \frac{6\pm 4}{2}}$;

${\displaystyle y_1=}$${\displaystyle \frac{6+4}{2}=}$${\displaystyle 5}$;

${\displaystyle y_2=}$${\displaystyle \frac{6-4}{2}=}$${\displaystyle 1}$;

6) ${\displaystyle z-3z^2=}$${\displaystyle -2}$;

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

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

${\displaystyle z_{1,2}=}$${\displaystyle \frac{1\pm \sqrt{25}}{2\cdot 3}=}$${\displaystyle \frac{1\pm 5}{6}}$;

${\displaystyle z_1=}$${\displaystyle \frac{1+5}{6}=}$${\displaystyle 1}$;

${\displaystyle z_2=}$${\displaystyle \frac{1-5}{6}=}$${\displaystyle \frac{-4}{6}=}$${\displaystyle -\frac{2}{3}}$;

7) ${\displaystyle 4x^2=}$${\displaystyle 2-7x}$;

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

${\displaystyle D=}$${\displaystyle 7^2-4\cdot 4\cdot (-2)=}$${\displaystyle 49+32=}$${\displaystyle 81}$;

${\displaystyle x_{1,2}=}$${\displaystyle \frac{-7\pm \sqrt{81}}{2\cdot 4}=}$${\displaystyle \frac{-7\pm 9}{8}}$;

${\displaystyle x_1=}$${\displaystyle \frac{-7+9}{8}=}$${\displaystyle \frac{2}{8}=}$${\displaystyle 0,25}$;

${\displaystyle x_2=}$${\displaystyle \frac{-7-9}{8}=}$${\displaystyle -2}$;

8) ${\displaystyle 4y^2=}$${\displaystyle 33+y}$;

${\displaystyle 4y^2-y-33=}$${\displaystyle 0}$;

${\displaystyle D=}$${\displaystyle {(-1)}^2-4\cdot 4\cdot (-33)=}$${\displaystyle 1+528=}$${\displaystyle 529}$;

${\displaystyle y_{1,2}=}$${\displaystyle \frac{1\pm \sqrt{529}}{2\cdot 4}=}$${\displaystyle \frac{1\pm 23}{8}}$;

${\displaystyle y_1=}$${\displaystyle \frac{1+23}{8}=}$${\displaystyle 3}$;

${\displaystyle y_2=}$${\displaystyle \frac{1-23}{8}=}$${\displaystyle \frac{-22}{8}=}$${\displaystyle -2,75}$;

9) ${\displaystyle 9x^2+25=}$${\displaystyle 30x}$;

${\displaystyle 9x^2-30x+25=}$${\displaystyle 0}$;

${\displaystyle D=}$${\displaystyle {(-30)}^2-4\cdot 9\cdot 25=}$${\displaystyle 900-900=}$${\displaystyle 0}$;

${\displaystyle x=}$${\displaystyle \frac{30}{18}=}$${\displaystyle 1\frac{2}{3}}$;

10) ${\displaystyle y^2=}$${\displaystyle 11y-18}$;

${\displaystyle y^2-11y+18=}$${\displaystyle 0}$;

${\displaystyle D=}$${\displaystyle {(-11)}^2-4\cdot 1\cdot 18=}$${\displaystyle 121-72=}$${\displaystyle 49}$;

${\displaystyle y_{1,2}=}$${\displaystyle \frac{11\pm \sqrt{49}}{2\cdot 1}=}$${\displaystyle \frac{11\pm 7}{2}}$;

${\displaystyle y_1=}$${\displaystyle \frac{11+7}{2}=}$${\displaystyle 9}$;

${\displaystyle y_2=}$${\displaystyle \frac{11-7}{2}=}$${\displaystyle 2}$;

11) ${\displaystyle 13z-3z^2=}$${\displaystyle 14}$;

${\displaystyle 3z^2-13z+14=}$${\displaystyle 0}$;

${\displaystyle D=}$${\displaystyle {(-13)}^2-4\cdot 3\cdot 14=}$${\displaystyle 169-168=}$${\displaystyle 1}$;

${\displaystyle z_{1,2}=}$${\displaystyle \frac{13\pm \sqrt{1}}{2\cdot 3}=}$${\displaystyle \frac{13\pm 1}{6}}$;

${\displaystyle z_1=}$${\displaystyle \frac{13+1}{6}=}$${\displaystyle \frac{14}{6}=}$${\displaystyle 2\frac{1}{3}}$;

${\displaystyle z_2=}$${\displaystyle \frac{13-1}{6}=}$${\displaystyle 2}$;

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

${\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}$;

13) ${\displaystyle 81y^2+1=}$${\displaystyle 18y}$;

${\displaystyle 81y^2-18y+1=}$${\displaystyle 0}$;

${\displaystyle D=}$${\displaystyle {(-18)}^2-4\cdot 81\cdot 1=}$${\displaystyle 324-324=}$${\displaystyle 0}$;

${\displaystyle y=}$${\displaystyle \frac{18}{162}=}$${\displaystyle \frac{1}{9}}$;

14) ${\displaystyle 6z-3=}$${\displaystyle z^2}$;

${\displaystyle z^2-6z+3=}$${\displaystyle 0}$;

${\displaystyle D=}$${\displaystyle {(-6)}^2-4\cdot 1\cdot 3=}$${\displaystyle 36-12=}$${\displaystyle 24}$;

${\displaystyle z_{1,2}=}$${\displaystyle \frac{6\pm \sqrt{24}}{2\cdot 1}=}$${\displaystyle \frac{6\pm 2\sqrt{6}}{2}=}$${\displaystyle 3\pm \sqrt{6}}$;

${\displaystyle z_1=}$${\displaystyle 3+\sqrt{6}}$;

${\displaystyle z_2=}$${\displaystyle 3-\sqrt{6}}$;

15) ${\displaystyle 18-y^2=}$${\displaystyle 5y+18}$;

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

${\displaystyle y(y+5)=}$${\displaystyle 0}$;

${\displaystyle y=}$${\displaystyle 0}$ або ${\displaystyle y+5=}$${\displaystyle 0}$;

${\displaystyle y_1=}$${\displaystyle 0}$ або ${\displaystyle y_2=}$${\displaystyle -5}$.