Вправи: 1070

1070. Розв’яжіть рівняння:

1) x2- 13x+1 + x+1x2-13 = 2,5.

Заміна x2- 13x+1 = t, тоді x+1x2- 13 = 1t.

t + 1t = 52; t + 1t – 52 = 0; 2t2- 5t+22t = 0.

D = (–5)² – 4 • 2 • 2 = 9; t₁ = 5+34 = 2; t₂ = 5-34 = 12.

1) t₁ = 2; x2- 13x+1 = 2; x² – 13 = 2x + 2,

x ≠ –1.

x² – 2x – 15 = 0; D = (–2)² – 4 • (–15) = 64;

x₁ = 2+8 2 = 5; x₂ = 2-82 = –3.

2) t₂ = 12; x2- 13x+1 = 12; 2x² – 26 = x + 1,

x ≠ –1.

2x² – x – 27 = 0; D = (–1)² – 4 • 2 • (–27) = 217; x_3,4 = 1 ± 2174.

Відповідь: –3; 5; 1 ± 2174.

2) x2+ 3x1-x + 5x-53x+ x2 = 4; x2+ 3x1-x – 5 • 1-x3x+ x2 = 4.

Заміна x2+ 3x1-x = t. Тоді t – 5t – 4 = 0; t2- 4t-5t = 0.

D = (–4)² – 4 • (–5) = 36; t₁ = 4+62 = 5; t₂ = 4-62 = –1.

1) t₁ = 5; x2+ 3x1-x = 5; x² + 3x = 5 – 5x,

x ≠ 1.

x² + 8x – 5 = 0; D = 8² – 4 • (–5) = 84; D = 221;

x_1,2 = -8 ± 2212 = 2(-4 ± 21)2 = –4 ± 21.

2) t₁ = –1; x2+ 3x1-x = –1; x² + 3x = –1 + x,

x ≠ 1.

x² + 2x + 1 = 0; (x + 1)² = 0; x = –1.

Відповідь: –4 ± 21; –1.