Алгебра 9 класс учебник Макарычев, Миндюк ответы – номер 501

2y² + 2y – 12 = 0
y² + y – 6 = 0

По теореме Виета
y₁ = 2, y₂ = – 3

1)x² + 2² + x + 2 = 18
x² + x + 6 – 18 = 0
x² + x – 12 = 0

По теореме Виета
x₁ = 3, x₂ = – 4

2)x² + (–3)² + x + (–3) = 18
x² + x + 9 – 3 – 18 = 0
x² + x – 12 = 0

По теореме Виета
x₁ = 3, x₂ = –4

Ответ: (3; 2), (–4; 2), (3; –3), (–4; –3)

Пусть xy = t, тогда x²y² = t²
t² + t = 72
t² + t – 72 = 0

По теореме Виета
t₁ = 8, t₂ = –9

6y – y² – 8 = 0
y² – 6y + 8 = 0

По теореме Виета
y₁ = 2, y₂ = 4
x₁ = 6 – 2 = 4
x₂ = 6 – 4 = 2

6y – y² + 9 = 0
y² – 6y – 9 = 0
D = (–6)² – 4 · 1 · (–9) = 72

у = $$\frac{6 ± \sqrt{72}}{2 \cdot 1}$$ = $$\frac{6 ± 6\sqrt{2}}{2}$$ = 3 ± 3$$\sqrt{2}$$

y₁ = 3 + 3$$\sqrt{2}$$, y₂ = 3 – $$\sqrt{2}$$

x₁ = 6 – (3 + 3$$\sqrt{2}$$) = 6 – 3 – 3$$\sqrt{2}$$ = 3 – 3$$\sqrt{2}$$

Ответ: (4; 2), (2; 4), (3 – 3$$\sqrt{2}$$; 3 + 3$$\sqrt{2}$$), (3 + 3$$\sqrt{2}$$; 3 – 3$$\sqrt{2}$$)

Пусть x + y = t
t² – 2t = 15
t² – 2t – 15 = 0

По теореме Виета
t₁ = – 3, t₂ = 5

–3 – y – 3y – y² + y – 11 = 0
–y² – 3y – 14 = 0
y² + 3y + 14 = 0
D = 3² – 4 · 1 · 14 = 9 – 56 = –47 < 0
Корней нет

5 – y + 5y – y² + y – 11 = 0
–y² + 5y – 6 = 0
y² – 5y + 6 = 0

По теореме Виета
y₁ = 2, y₂ = 3
x₁ = 5 – 2 = 3
x₂ = 5 – 3 = 2

Ответ: (3; 2), (2; 3)

Пусть x – y = t
t² – 2t = 3
t² – 2t – 3 = 0

По теореме Виета
t₁ = – 1, t₂ = 3

(2y – 1)² – 4(2y – 1) = 45
4y² – 4y + 1 – 8y + 4 – 45 = 0
4y² – 12y – 40 = 0
y² – 3y – 10 = 0

По теореме Виета
y₁ = – 2, y₂ = 5
x₁ = – 2 – 1 = –3
x₂ = 5 – 1 = 4

(2y + 3)² – 4(2y + 3) = 45
4y² + 12y + 9 – 8y – 12 – 45 = 0
4y² + 4y – 48 = 0
y² + y – 12 = 0

По теореме Виета
y₁ = –4, y₂ = 3
x₁ = –4 + 3 = –1
x₂ = 3 + 3 = 6

Ответ: (–3; –2), (4; 5), (–1; –4), (6; 3)