Алгебра 9 класс учебник Макарычев, Миндюк ответы – номер 852

x³ – y³ = 447 + x²y – xy²
x³ – y³ = 447 + 210
x³ – y³ = 657



$$\frac{657}{х^2 + xy + y^2}$$ = $$\frac{210}{xy}$$

657xy = 210(x² + xy + y²)
657xy = 210x² + 210xy + 210y²
657 = 210 · x/y + 210 + 210 · y/x
Пусть x/y = m
657 = 210m + 210 + 210 · 1/m
210m + 210 · 1/m – 447 = 0
210m² + 210 – 447m = 0
210m² – 447m + 210 = 0
70m² – 149m + 70 = 0
D₁ = (–149)² – 4 · 70 · 70 = 22201 – 19600 = 2601

m = $$\frac{149 ± \sqrt{2601}}{2 \cdot 70}$$ = $$\frac{149 ± \sqrt{51}}{140}$$

m₁ = 200/140 = 10/7

m₂ = 98/140 = 7/10

1) x/y = 10/7
7x = 10y
y = 0,7x
x · 0,7x · (x – 0,7x) = 210
0,7x² · 0,3x = 210
0,21x³ = 210
x³ = 1000
x = 10
y = 0,7 · 10 = 7

2) x/y = 7/10
10x = 7y
x = 0,7y
0,7y · y(0,7y – y) = 210
0,7y² · (–0,3y) = 210
–0,21y³ = 210
y³ = –1000
y = –10
x = 0,7 · (–10) = –7

Ответ: (10; 7) и (–7; 10)

$$\frac{30}{xy}$$ = $$\frac{35}{x^2 - xy + y^2}$$

35xy = 30(x² – xy + y²)
35xy = 30x² – 30xy + 30y²
35 = 30 · x/y – 30 + 30 · y/x
Пусть x/y = m
35 = 30m – 30 + 30 · 1/m
30m + 30 · 1/m – 65 = 0
30m² + 30 – 65m = 0
6m² – 13m + 6 = 0
D₁ = (–13)² – 4 · 6 · 6 = 169 – 144 = 25

m = $$\frac{13 ± \sqrt{25}}{2 \cdot 6}$$ = $$\frac{13 ± 5}{12}$$

m₁ = 18/12 = 3/2

m₂ = 8/12 = 2/3

1) x/y = 3/2
2x = 3y
x = 1,5y
1,5y · y(1,5y + y) = 30
1,5y² · 2,5y = 30
3,75y³ = 30
y³ = 8
y = 2
x = 1,5 · 2 = 3

2) x/y = 2/3
2y = 3x
y = 1,5x
x · 1,5x · (x + 1,5x) = 30
1,5x² · 2,5x = 30
3,75x³ = 30
x³ = 8
x = 2
y = 1,5 · 2 = 3

Ответ: (2; 3) и (3; 2)