760 Вправа 760Алгебра 8 клас Тарасенкова 2025 · § 18 · стор. 226

760. Розв’яжіть рівняння:

1) ${\displaystyle 2x^2=}$${\displaystyle 7x+30}$;

${\displaystyle 2x^2-7x-30=}$${\displaystyle 0}$;

${\displaystyle D=}$${\displaystyle {(-7)}^2-4\cdot 2\cdot (-30)=}$${\displaystyle 49+240=}$${\displaystyle 289}$;

${\displaystyle x_{1,2}=}$${\displaystyle \frac{7\pm \sqrt{289}}{2\cdot 2}=}$${\displaystyle \frac{7\pm 17}{4}}$;

${\displaystyle x_1=}$${\displaystyle \frac{7+17}{4}=}$${\displaystyle 6}$;

${\displaystyle x_2=}$${\displaystyle \frac{7-17}{4}=}$${\displaystyle \frac{-10}{4}=}$${\displaystyle -2,5}$;

2) ${\displaystyle 3y=}$${\displaystyle 2y^2-5}$;

${\displaystyle 2y^2-3y-5=}$${\displaystyle 0}$;

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

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

${\displaystyle y_1=}$${\displaystyle \frac{3+7}{4}=}$${\displaystyle \frac{10}{4}=}$${\displaystyle 2,5}$;

${\displaystyle y_2=}$${\displaystyle \frac{3-7}{4}=}$${\displaystyle -1}$;

3) ${\displaystyle z^2-5=}$${\displaystyle 2z}$;

${\displaystyle z^2-2z-5=}$${\displaystyle 0}$;

${\displaystyle D=}$${\displaystyle {(-2)}^2-4\cdot 1\cdot (-5)=}$${\displaystyle 4+20=}$${\displaystyle 24}$;

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

${\displaystyle z_1=}$${\displaystyle 1+\sqrt{6}}$;

${\displaystyle z_2=}$${\displaystyle 1-\sqrt{6}}$;

4) ${\displaystyle 30-x=}$${\displaystyle x^2}$;

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

${\displaystyle D=}$${\displaystyle 1^2-4\cdot 1\cdot (-30)=}$${\displaystyle 1+120=}$${\displaystyle 121}$;

${\displaystyle x_{1,2}=}$${\displaystyle \frac{-1\pm \sqrt{121}}{2\cdot 1}=}$${\displaystyle \frac{-1\pm 11}{2}}$;

${\displaystyle x_1=}$${\displaystyle \frac{-1+11}{2}=}$${\displaystyle 5}$;

${\displaystyle x_2=}$${\displaystyle \frac{-1-11}{2}=}$${\displaystyle -6}$;

5) ${\displaystyle 5y=}$${\displaystyle 1-14y^2}$;

${\displaystyle 14y^2+5y-1=}$${\displaystyle 0}$;

${\displaystyle D=}$${\displaystyle 5^2-4\cdot 14\cdot (-1)=}$${\displaystyle 25+56=}$${\displaystyle 81}$;

${\displaystyle y_{1,2}=}$${\displaystyle \frac{-5\pm \sqrt{81}}{2\cdot 14}=}$${\displaystyle \frac{-5\pm 9}{28}}$;

${\displaystyle y_1=}$${\displaystyle \frac{-5+9}{28}=}$${\displaystyle \frac{4}{28}=}$${\displaystyle \frac{1}{7}}$;

${\displaystyle y_2=}$${\displaystyle \frac{-5-9}{28}=}$${\displaystyle \frac{-14}{28}=}$${\displaystyle -0,5}$;

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

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

${\displaystyle 4x(x+3)=}$${\displaystyle 0}$;

${\displaystyle x=}$${\displaystyle 0}$ або ${\displaystyle x+3=}$${\displaystyle 0}$;

${\displaystyle x_1=}$${\displaystyle 0}$ або ${\displaystyle x_2=}$${\displaystyle -3}$.