1035 Вправа 1035Алгебра 8 клас Бевз 2025 · § 20 · стор. 196

1035. Склади зведене квадратне рівняння, корені якого дорівнюють: а) 2 і 4; б) −3 і 5; в) 0,5 і 4; г) 1/7 і 7; ґ) 2 − √3 і 2 + √3.

Зведене рівняння ${\displaystyle x^2+px+q=}$${\displaystyle 0}$, де ${\displaystyle p=}$${\displaystyle -(x_1+x_2)}$;

${\displaystyle q=}$${\displaystyle x_1{x}_2}$.

а) ${\displaystyle x_1=}$${\displaystyle 2}$;

${\displaystyle x_2=}$${\displaystyle 4}$;

${\displaystyle p=}$${\displaystyle -(2+4)=}$${\displaystyle -6}$;

${\displaystyle q=}$${\displaystyle 2\cdot 4=}$${\displaystyle 8}$;

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

б) ${\displaystyle x_1=}$${\displaystyle -3}$;

${\displaystyle x_2=}$${\displaystyle 5}$;

${\displaystyle p=}$${\displaystyle -(-3+5)=}$${\displaystyle -2}$;

${\displaystyle q=}$${\displaystyle -3\cdot 5=}$${\displaystyle -15}$;

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

в) ${\displaystyle x_1=}$${\displaystyle 0,5}$;

${\displaystyle x_2=}$${\displaystyle 4}$;

${\displaystyle p=}$${\displaystyle -(0,5+4)=}$${\displaystyle -4,5}$;

${\displaystyle q=}$${\displaystyle 0,5\cdot 4=}$${\displaystyle 2}$;

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

г) ${\displaystyle x_1=}$${\displaystyle \frac{1}{7}}$;

${\displaystyle x_2=}$${\displaystyle 7}$;

${\displaystyle p=}$${\displaystyle -(\frac{1}{7}+7)=}$${\displaystyle -\frac{50}{7}}$;

${\displaystyle q=}$${\displaystyle \frac{1}{7}\cdot 7=}$${\displaystyle 1}$;

${\displaystyle x^2-\frac{50}{7}x+1=}$${\displaystyle 0}$;

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

ґ) ${\displaystyle x_1=}$${\displaystyle 2-\sqrt{3}}$;

${\displaystyle x_2=}$${\displaystyle 2+\sqrt{3}}$;

${\displaystyle x_1+x_2=}$${\displaystyle 4}$;

${\displaystyle x_1{x}_2=}$${\displaystyle 2^2-{\left(\sqrt{3}\right)}^2=}$${\displaystyle 4-3=}$${\displaystyle 1}$;

${\displaystyle x^2-4x+1=}$${\displaystyle 0}$.