2.86.
x⁴ - 2x³ - 18x² - 6x + 9 = 0
Видим, что x = -1 является корнем уравнения:
1 + 2 - 18 + 6 + 9 = 0
x⁴ + x³ - 3x³ - 3x² - 15x² - 15x + 9x + 9 = 0
x³(x + 1) - 3x²(x + 1) - 15x(x + 1) + 9(x + 1) = 0
(x³ - 3x² - 15x + 9)(x + 1) = 0
x = -1
x³ - 3x² - 15x + 9 = 0
x³ + 3x² - 6x² - 18x + 3x + 9 = 0
x²(x + 3) - 6x(x + 3) + 3(x + 3) = 0
(x + 3)(x² - 6x + 3) = 0
x = -3
x² - 6x + 3 = 0
x² - 6x + 9 - 6 = 0
(x - 3)² - (√6)² = 0
(x - 3 - √6)(x - 3 + √6) = 0
x = 3 + √6; 3 - √6
Ответ: x = -3; -1; 3 - √6; 3 + √6.
2.87.
4x⁴ - 8x³ + 3x² - 8x + 4 = 0
Это возвратное уравнение. Делим на x².
4x² - 8x + 3 - 8/x + 4/x² = 0
4x² + 4/x² + 3 - 8x - 8/x = 0
4x² + 8 + 4/x² - 8x - 8/x - 5 = 0
4(x² + 2 + 1/x²) - 8(x + 1/x) - 5 = 0
4(x + 1/x)² - 8(x + 1/x) - 5 = 0
Пусть x + 1/x = t.
4t² - 8t - 5 = 0
D = 64 + 5·4·4 = 16·4 + 16·5 = 16·9 = 12²
t₁ = (8 + 12)/8 = 20/8 = 5/2
t₂ = (8 - 12)/8 = -4/8 = -1/2
Обратная замена:
1) x + 1/x = 5/2
2x² + 2 = 5x
2x² - 5x + 2 = 0
2x² - 4x - x + 2 = 0
2x(x - 2) - (x - 2) = 0
(2x - 1)(x - 2) = 0
x = 1/2; 2.
2) x + 1/x = -1/2
2x² + 2 = -x
2x² + x + 2 = 0
D = 1 - 2·2·4 < 0 ⇒ нет корней
Ответ: x = 1/2; 2.
2.88
3x² + 5x + 5/x + 3/x² = 16
3x² + 6 + 3/x² + 5x + 5/x - 16 - 6 = 0
3(x² + 2 + 1/x²) + 5(x + 1/x) - 22 = 0
3(x + 1/x)² + 5(x + 1/x) - 22 = 0
Пусть t = 1 + 1/x.
3t² + 5t - 22 = 0
D = 25 + 22·4·3 = 289 = 17²
t₁ = (-5 + 17)/6 = 12/6 = 2
t₂ = (-5 - 17)/6 = -22/6 = -11/3
Обратная замена:
x + 1/x = 2
x² + 1 = 2x
x² - 2x + 1 = 0
(x - 1)² = 0
x = 1
2) x + 1/x = -11/3
3x² + 3 = -11x
3x² + 11x + 3 = 0
D = 11² - 3·3·4 = 121 - 36 = 85 = (√85)²
x₁ = (-11 + √85)/6
x₂ = (-11 - √85)/6
Ответ: x = (-11 - √85)/6; (-11 + √85)/6; 1.
2.89.
x⁴ - 2x³ - 13x² - 2x + 1 = 0
Тоже возвратное уравнение.
Разделим на x².
x² - 2x - 13 - 2/x + 1/x² = 0
x² + 1/x² - 2x - 2/x - 13 = 0
x² + 2 + 1/x² - 2x - 2/x - 13 - 2 = 0
(x + 1/x)² - 2(x + 1/x) - 15 = 0
Пусть t = 1 + 1/x.
t² - 2t - 15 = 0
t² - 2t + 1 - 16 = 0
(t - 1)² - 4² = 0
(t - 1 - 4)(t - 1 + 4) = 0
(t - 5)(t + 3) = 0
t = -3; 5
Обратная замена:
1) x + 1/x = -3
x² + 1 = -3x
x² + 3x + 1 = 0
D = 9 - 4 = 5 = (√5)²
x₁ = (-3 + √5)/2
x₂ = (-3 - √5)/2
2) x + 1/x = 5
x² + 1 = 5x
x² - 5x + 1 = 0
D = 25 - 4 = 21 = (√21)²
x₁ = (5 + √21)/2
x₂ = (5 - √21)/2
Ответ: x = (-3 - √5)/2; (-3 + √5)/2; (5 - √21)/2; (5 + √21)/2.