1293 Вправа 1293Алгебра 8 клас Бевз 2025 · § 26 · стор. 251

1293. Розв’яжи математичні кросворди (мал. 26.5).

Лівий. Позначимо кружечки: верхній лівий ${\displaystyle a}$, верхній правий ${\displaystyle b}$, нижній лівий ${\displaystyle c}$, нижній правий ${\displaystyle d}$.

Записи: ${\displaystyle 45:a=}$${\displaystyle b}$;

${\displaystyle a\cdot 4=}$${\displaystyle c}$;

${\displaystyle b+3=}$${\displaystyle d}$;

${\displaystyle c-28=}$${\displaystyle d}$;

тоді ${\displaystyle \frac{45}{a}+3=}$${\displaystyle 4a-28}$:

${\displaystyle \frac{45}{a}+3=}$${\displaystyle 4a-28\mid \cdot a}$;

${\displaystyle 45+3a=}$${\displaystyle 4a^2-28a}$;

${\displaystyle 4a^2-31a-45=}$${\displaystyle 0}$;

${\displaystyle a=}$${\displaystyle 4,b=}$${\displaystyle -31,c=}$${\displaystyle -45}$;

${\displaystyle D=}$${\displaystyle b^2-4ac=}$${\displaystyle {(-31)}^2-4\cdot 4\cdot (-45)=}$${\displaystyle 961+720=}$${\displaystyle 1681}$;

${\displaystyle D>0}$, два корені:

${\displaystyle a_{1,2}=}$${\displaystyle \frac{-b\pm \sqrt{D}}{2a}=}$${\displaystyle \frac{31\pm \sqrt{1681}}{8}=}$${\displaystyle \frac{31\pm 41}{8}}$;

${\displaystyle a_1=}$${\displaystyle \frac{31+41}{8}=}$${\displaystyle 9}$;

${\displaystyle a_2=}$${\displaystyle \frac{31-41}{8}=}$${\displaystyle -1,25}$;

у кружечки вписуємо натуральні числа, тому ${\displaystyle a=}$${\displaystyle -1,25}$ не підходить;

${\displaystyle a=}$${\displaystyle 9}$: ${\displaystyle 45:9=}$${\displaystyle 5}$;

${\displaystyle 9\cdot 4=}$${\displaystyle 36}$;

${\displaystyle 5+3=}$${\displaystyle 8}$;

${\displaystyle 36-28=}$${\displaystyle 8}$;

Правий. Записи: ${\displaystyle 45:a=}$${\displaystyle b}$;

${\displaystyle 8+a=}$${\displaystyle c}$;

${\displaystyle 18+b=}$${\displaystyle d}$;

${\displaystyle c\cdot 3=}$${\displaystyle d}$;

тоді ${\displaystyle 3(8+a)=}$${\displaystyle 18+\frac{45}{a}}$:

${\displaystyle 3(8+a)=}$${\displaystyle 18+\frac{45}{a}\mid \cdot a}$;

${\displaystyle 24a+3a^2=}$${\displaystyle 18a+45}$;

${\displaystyle 3a^2+6a-45=}$${\displaystyle 0\mid :3}$;

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

${\displaystyle a=}$${\displaystyle 1,b=}$${\displaystyle 2,c=}$${\displaystyle -15}$;

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

${\displaystyle D>0}$, два корені:

${\displaystyle a_{1,2}=}$${\displaystyle \frac{-b\pm \sqrt{D}}{2a}=}$${\displaystyle \frac{-2\pm \sqrt{64}}{2}=}$${\displaystyle \frac{-2\pm 8}{2}}$;

${\displaystyle a_1=}$${\displaystyle \frac{-2+8}{2}=}$${\displaystyle 3}$;

${\displaystyle a_2=}$${\displaystyle \frac{-2-8}{2}=}$${\displaystyle -5}$;

${\displaystyle a=}$${\displaystyle -5}$ — від’ємне, беремо натуральне ${\displaystyle a=}$${\displaystyle 3}$;

${\displaystyle a=}$${\displaystyle 3}$: ${\displaystyle 45:3=}$${\displaystyle 15}$;

${\displaystyle 8+3=}$${\displaystyle 11}$;

${\displaystyle 18+15=}$${\displaystyle 33}$;

${\displaystyle 11\cdot 3=}$${\displaystyle 33}$.