Вправи: 1069 (1-2)

1069. Знайдіть корені рівняння:

1) (х² – 4х)(х – 2)² + 3 = 0; (х² – 4х)(х² – 4х + 4) + 3 = 0.

Заміна x² – 4х = t . Тоді t(t + 4) + 3 = 0; t² + 4t + 3 = 0;

D = 4² – 4 • 3 = 4;

t₁ = -4+22 = –1; t₂ = -4-22 = –3.

1) t₁ = –1; x² – 4x = –1; x² – 4x + 1 = 0; D = (–4)² – 4 • 1 = 12;

x_1,2 = 4 ±232 = 2(2 ± 3)2 = 2 ± 3.

2) t₂ = –3; x² – 4х = –3; х² – 4х + 3 = 0; D = (–4)² – 4 • 3 = 4;

x₃ = 4+22 = 3; x₄ = 4-22 = 1.

Відповідь: 2 ± 3; 1; 3.

2) х(х – 1)(х – 2)(х – 3) = 24; (х(х – 3))((х – 1)(х – 2)) = 24;

(х² – Зх)(х² – Зх + 2) = 24.

Заміна x² – Зх = t . Маємо t(t + 2) – 24 = 0; t² + 2t – 24 = 0;

D = 2² – 4 • (–24) = 100; t₁ = -2+102 = 4; t₂ = -2-102 = –6.

1) t₁ = 4; x² – Зх = 4; х² – Зх – 4 – 0; D = (–3)² – 4 • (–4) = 25;

x₁ = 3+52 = 4; x₂ = 3-52 = –1.

2) t₂ = –6; x² – Зх = –6; х² – Зх + 6 = 0; D = (–3)² – 4 • 6 < 0, немає розв’язків.

Відповідь: –1; 4.

3) x² – 3x = 8x2- 3x-2.

Заміна x² – 3x = t; t = 8t-2; t² – 2t = 8,

t ≠ 2.

t² – 2t – 8 = 0; D = (–2)² – 4 • (–8) = 36;

t₁ = 2+62 = 4; t₂ = 2-62 = –2.

1) t = 4; x² – Зх = 4; x² – Зх – 4 = 0; D = (–3)² – 4 • (–4) = 25;

x₁ = 3+52 = 4; x₂ = 3-52 = –1.

2) t = –2; x² – Зх = –2; х² – Зх + 2 = 0; D = (–3)² – 4 • 2 = 1;

x₃ = 3+12 = 2; x₄ = 3-12 = 1.

Відповідь: –1; 1; 2; 4.

4) (x + 2)(x – 7) = 19x-1(x-4); x² – 5x – 14 = 19x2- 5x+4.

Заміна x² – 5x = t, тоді t – 14 = 19t+4.

t² – 14t + 4t – 56 = 19, t² – 10t – 75 = 0,

t ≠ –4; t ≠ – 4.

D = (–10)² – 4 • (–75) = 400; t₁ = 10+202 = 15; t₂ = 10-202 = –5.

1) t₁ = 15; x² – 5x = 15; x² – 5x – 15 = 0; D = (–5)² – 4 • (–15) = 85; x_1,2 = 5 ± 852.

2) t₂ = –5; x² – 5x = –5; x² – 5x + 5 = 0; D = (–5)² – 4 • 5 = 5;

x_3,4 = 5 ± 52

Відповідь: 5 ± 852; 5 ± 52.

5) 5x2- x-x + 1x2- x-5 = 2.

Заміна x² – x – 1 = t. Tоді 5t + 1t-4 = 2; 5t-20+tt(t-4) = 2;

2t(t – 4) = 5t – 20;

t ≠ 0,

t ≠ 4.

2t² – 8t – 6t + 20 = 0; 2t² – 14t + 20 = 0; t² – 7t + 10 = 0;

D = (–7)² – 4 • 10 = 9;

t₁ = 7+32 = 5; t₂ = 7-32 = 2.

1) t₁ = 5; x² – x – 1 = 5; x² – x – 6 = 0; D = (–1)² – 4 • (–6) = 25;

x₁ = 1+52 = 3; x₂ = 1-52 = –2.

2) t₂ = 2; x² – x – 1 = 2; x² – x – 3 = 0; D = (–1)² – 4 • (–3) = 13;

x_3,4 = 1 ± 132.

Відповідь: 3; –2; 1 ± 132.

6) 2x2- 11x+4 + 3x2- 11x+1 = 8x2- 11x-2.

Заміна x² – 11x = t. Тоді 2t+4 + 3t+1 – 8t-2 = 0;

ОДЗ: t ≠ –4; t ≠ –1; t ≠ 2.

2(t + 1)(t – 2) + 3(t + 4)(t – 2) – 8(t + 4)(t + 1) = 0;

2t² + 2t – 4t – 4 + 3t² + 12t – 6t – 24 – 8t² – 32t – 8t – 32 = 0;

–3t² – 36t – 60 = 0; t² + 12t + 20 = 0; D = 12² – 4 • 20 = 64;

t₁ = -12+82 = –2; t₂ = -12-82 = –10.

1) t₁ = –2; x² – 11x = –2; x² – 11x + 2 = 0; D = (–11)² – 4 = 113;

x_1,2 = 11 ± 1132.

2) t₂ = –10; x² – 11x = –10; x² – 11x + 10 = 0;

D = (–11)² – 4 • 10 = 81;

x₃ = 11+92 = 10; x₄ = 11-92 = 1.

Відповідь: 11 ± 1132; 10; 1.