755° Вправа 755Алгебра 8 клас Тарасенкова 2025 · § 18 · стор. 225

755. Розв’яжіть квадратне рівняння:

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

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

${\displaystyle x_{1,2}=}$${\displaystyle \frac{-4\pm \sqrt{36}}{2\cdot 1}=}$${\displaystyle \frac{-4\pm 6}{2}}$;

${\displaystyle x_1=}$${\displaystyle \frac{-4+6}{2}=}$${\displaystyle 1}$;

${\displaystyle x_2=}$${\displaystyle \frac{-4-6}{2}=}$${\displaystyle -5}$;

2) ${\displaystyle x^2-6x-16=}$${\displaystyle 0}$;

${\displaystyle D=}$${\displaystyle {(-6)}^2-4\cdot 1\cdot (-16)=}$${\displaystyle 36+64=}$${\displaystyle 100}$;

${\displaystyle x_{1,2}=}$${\displaystyle \frac{6\pm \sqrt{100}}{2\cdot 1}=}$${\displaystyle \frac{6\pm 10}{2}}$;

${\displaystyle x_1=}$${\displaystyle \frac{6+10}{2}=}$${\displaystyle 8}$;

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

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

${\displaystyle D=}$${\displaystyle 2^2-4\cdot 1\cdot (-8)=}$${\displaystyle 4+32=}$${\displaystyle 36}$;

${\displaystyle x_{1,2}=}$${\displaystyle \frac{-2\pm \sqrt{36}}{2\cdot 1}=}$${\displaystyle \frac{-2\pm 6}{2}}$;

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

${\displaystyle x_2=}$${\displaystyle \frac{-2-6}{2}=}$${\displaystyle -4}$;

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

${\displaystyle D=}$${\displaystyle {(-8)}^2-4\cdot 1\cdot 16=}$${\displaystyle 64-64=}$${\displaystyle 0}$;

${\displaystyle x=}$${\displaystyle \frac{8}{2}=}$${\displaystyle 4}$;

5) ${\displaystyle x^2+8x+7=}$${\displaystyle 0}$;

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

${\displaystyle x_{1,2}=}$${\displaystyle \frac{-8\pm \sqrt{36}}{2\cdot 1}=}$${\displaystyle \frac{-8\pm 6}{2}}$;

${\displaystyle x_1=}$${\displaystyle \frac{-8+6}{2}=}$${\displaystyle -1}$;

${\displaystyle x_2=}$${\displaystyle \frac{-8-6}{2}=}$${\displaystyle -7}$;

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

${\displaystyle D=}$${\displaystyle {(-4)}^2-4\cdot 1\cdot 8=}$${\displaystyle 16-32=}$${\displaystyle -16}$;

${\displaystyle D<0}$ — коренів немає;

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

${\displaystyle D=}$${\displaystyle 1^2-4\cdot 1\cdot (-12)=}$${\displaystyle 1+48=}$${\displaystyle 49}$;

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

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

${\displaystyle x_2=}$${\displaystyle \frac{-1-7}{2}=}$${\displaystyle -4}$;

8) ${\displaystyle x^2-2x-15=}$${\displaystyle 0}$;

${\displaystyle D=}$${\displaystyle {(-2)}^2-4\cdot 1\cdot (-15)=}$${\displaystyle 4+60=}$${\displaystyle 64}$;

${\displaystyle x_{1,2}=}$${\displaystyle \frac{2\pm \sqrt{64}}{2\cdot 1}=}$${\displaystyle \frac{2\pm 8}{2}}$;

${\displaystyle x_1=}$${\displaystyle \frac{2+8}{2}=}$${\displaystyle 5}$;

${\displaystyle x_2=}$${\displaystyle \frac{2-8}{2}=}$${\displaystyle -3}$;

9) ${\displaystyle x^2-7x-8=}$${\displaystyle 0}$;

${\displaystyle D=}$${\displaystyle {(-7)}^2-4\cdot 1\cdot (-8)=}$${\displaystyle 49+32=}$${\displaystyle 81}$;

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

${\displaystyle x_1=}$${\displaystyle \frac{7+9}{2}=}$${\displaystyle 8}$;

${\displaystyle x_2=}$${\displaystyle \frac{7-9}{2}=}$${\displaystyle -1}$;

10) ${\displaystyle x^2+6x+3=}$${\displaystyle 0}$;

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

${\displaystyle x_{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 x_1=}$${\displaystyle -3+\sqrt{6}}$;

${\displaystyle x_2=}$${\displaystyle -3-\sqrt{6}}$;

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

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

${\displaystyle x_{1,2}=}$${\displaystyle \frac{-1\pm \sqrt{9}}{2\cdot 1}=}$${\displaystyle \frac{-1\pm 3}{2}}$;

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

${\displaystyle x_2=}$${\displaystyle \frac{-1-3}{2}=}$${\displaystyle -2}$;

12) ${\displaystyle x^2+25x+100=}$${\displaystyle 0}$;

${\displaystyle D=}$${\displaystyle {25}^2-4\cdot 1\cdot 100=}$${\displaystyle 625-400=}$${\displaystyle 225}$;

${\displaystyle x_{1,2}=}$${\displaystyle \frac{-25\pm \sqrt{225}}{2\cdot 1}=}$${\displaystyle \frac{-25\pm 15}{2}}$;

${\displaystyle x_1=}$${\displaystyle \frac{-25+15}{2}=}$${\displaystyle -5}$;

${\displaystyle x_2=}$${\displaystyle \frac{-25-15}{2}=}$${\displaystyle -20}$;

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

${\displaystyle D=}$${\displaystyle {(-6)}^2-4\cdot 1\cdot (-7)=}$${\displaystyle 36+28=}$${\displaystyle 64}$;

${\displaystyle x_{1,2}=}$${\displaystyle \frac{6\pm \sqrt{64}}{2\cdot 1}=}$${\displaystyle \frac{6\pm 8}{2}}$;

${\displaystyle x_1=}$${\displaystyle \frac{6+8}{2}=}$${\displaystyle 7}$;

${\displaystyle x_2=}$${\displaystyle \frac{6-8}{2}=}$${\displaystyle -1}$;

14) ${\displaystyle x^2+15x+26=}$${\displaystyle 0}$;

${\displaystyle D=}$${\displaystyle {15}^2-4\cdot 1\cdot 26=}$${\displaystyle 225-104=}$${\displaystyle 121}$;

${\displaystyle x_{1,2}=}$${\displaystyle \frac{-15\pm \sqrt{121}}{2\cdot 1}=}$${\displaystyle \frac{-15\pm 11}{2}}$;

${\displaystyle x_1=}$${\displaystyle \frac{-15+11}{2}=}$${\displaystyle -2}$;

${\displaystyle x_2=}$${\displaystyle \frac{-15-11}{2}=}$${\displaystyle -13}$;

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

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

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

${\displaystyle x_1=}$${\displaystyle \frac{3+1}{2}=}$${\displaystyle 2}$;

${\displaystyle x_2=}$${\displaystyle \frac{3-1}{2}=}$${\displaystyle 1}$;

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

${\displaystyle D=}$${\displaystyle {(-10)}^2-4\cdot 1\cdot 25=}$${\displaystyle 100-100=}$${\displaystyle 0}$;

${\displaystyle x=}$${\displaystyle \frac{10}{2}=}$${\displaystyle 5}$.