1136 Вправа 1136Алгебра 8 клас Бевз 2025 · § 22 · стор. 213

1136. A boat traveled 50 km downstream and 45 km upstream, taking 10 hours to complete the journey. Find the boat’s own speed if the river’s current is 2,5 km/h. Write the boat’s own velocity as x, create a table for the problem, and then write and solve the equation.

${\displaystyle \frac{50}{x+2,5}+\frac{45}{x-2,5}=}$${\displaystyle 10\mid \cdot \left(x+2,5\right)\left(x-2,5\right)}$;

ОДЗ: ${\displaystyle x>2,5}$;

${\displaystyle 50(x-2,5)+45(x+2,5)=}$${\displaystyle 10(x^2-6,25)}$;

${\displaystyle 95x-12,5=}$${\displaystyle 10x^2-62,5}$;

${\displaystyle 10x^2-95x-50=}$${\displaystyle 0\mid :5}$;

${\displaystyle 2x^2-19x-10=}$${\displaystyle 0}$;

${\displaystyle a=}$${\displaystyle 2,b=}$${\displaystyle -19,c=}$${\displaystyle -10}$;

${\displaystyle D=}$${\displaystyle b^2-4ac=}$${\displaystyle {(-19)}^2-4\cdot 2\cdot (-10)=}$${\displaystyle 361+80=}$${\displaystyle 441}$;

${\displaystyle D>0}$, два корені:

${\displaystyle x_{1,2}=}$${\displaystyle \frac{-b\pm \sqrt{D}}{2a}=}$${\displaystyle \frac{19\pm \sqrt{441}}{4}=}$${\displaystyle \frac{19\pm 21}{4}}$;

${\displaystyle x_1=}$${\displaystyle \frac{19+21}{4}=}$${\displaystyle 10}$;

${\displaystyle x_2=}$${\displaystyle \frac{19-21}{4}=}$${\displaystyle -0,5}$;

Отже, ${\displaystyle x=}$${\displaystyle 10\text{або }x=}$${\displaystyle -0,5}$;

${\displaystyle x=}$${\displaystyle -0,5}$ — не підходить.

Відповідь: 10 km/h.