Алгебра 9 класс учебник Макарычев, Миндюк ответы – номер 853

x² + (6/х)² = 13
x⁴ + 36 = 13x²
x⁴ – 13x² + 36 = 0
пусть x² = t
t² – 13t + 36 = 0

по теореме Виета
t₁ = 4; t₂ = 9
x² = 4
x₁ = 2
x₂ = –2
y₁ = 6/2 = 3
y₂ = 6/-2 = –3
или
x² = 9
x₃ = 3
x₄ = –3
y₃ = 6/3 = 2
y₄ = 6/-3 = –2

2) x = 0, y = 0

3) (x – y)(x² + xy + y²) = 19(x – y)
(x – y)(x² + xy + y²) – 19(x – y) = 0
(x – y)(x² + xy + y² – 19) = 0
x – y = 0
x = y
x² – xy + y² = 7
x² – x² + x² = 7
x² = 7
x₁ = $$\sqrt{7}$$, x₂ = –$$\sqrt{7}$$
y₁ = = $$\sqrt{7}$$, y₂ = –$$\sqrt{7}$$

4) (x + y)(x² – xy + y²) = 7(x + y)
(x + y)(x² – xy + y²) – 7(x + y) = 0
(x + y)(x² – xy + y² – 7) = 0
x = –y
x² + xy + y² = 19
y² + (–y) · y + y² = 19
y² = 19
y₁ = $$\sqrt{19}$$; y₂ = –$$\sqrt{19}$$
x₁ = –$$\sqrt{19}$$; x₂ = $$\sqrt{19}$$

Ответ: (2; 3), (–2; 3), (3; 2), (–3; 2), (0; 0), (–$$\sqrt{19}$$; $$\sqrt{19}$$); ($$\sqrt{19}$$; –$$\sqrt{19}$$); ($$\sqrt{7}$$; $$\sqrt{7}$$), (–$$\sqrt{7}$$; –$$\sqrt{7}$$)