1) (x² + x + 1)(x² + x + 2) = 12
Сделаем замену :
x² + x + 1 = m , тогда x² + x + 2 = m + 1
m(m + 1) = 12
m² + m - 12 = 0
D = 1² - 4 * (- 12) = 1 + 48 = 49 = 7²
1) x² + x + 1 = - 4
x² + x + 5 = 0
D = 1² - 4 * 5 = 1 - 20 = - 19 < 0 - решений нет
2) x² + x + 1 = 3
x² + x - 2 = 0
По теореме Виета :
x₁ = - 2 x₂ = 1
ответ : - 2 ; 1
2)
3(x² + 5x + 1)² + 2x² + 10x = 3
3(x² + 5x + 1)² + 2(x² + 5x) = 3
x² + 5x + 1 = m , тогда x² + 5x = m - 1
3m² + 2(m - 1) = 3
3m² + 2m - 2 - 3 = 0
3m² + 2m - 5 = 0
D = 2² - 4 * 3 * (- 5) = 4 + 60 = 64 = 8²
1)x²+ 5x + 1 = - 5/3
x² + 5x + 8/3 = 0
3x² + 15x + 8 = 0
D = 15² - 4 * 3 * 8 = 225 - 96 = 129
2)x²+ 5x + 1 = 1
x² + 5x = 0
x(x + 5) = 0
x₃ = 0 x₄ = - 5
3)
(x⁴ - 5x²)² - 2(x⁴ - 5x²) = 24
x⁴ - 5x² = m
m² - 2m - 24 = 0
m₁ = 6 m₂ = - 4
1) x⁴ - 5x² = 6
x⁴ - 5x² - 6 = 0
x² = 6
x₁ = - √6 x₂ = √6
x² = - 1 - решений нет
2) x⁴ - 5x² = - 4
x⁴ - 5x² + 4 = 0
x² = 4
x₃ = - 2 x₄ = 2
x² = 1
x₅ = - 1 x₆ = 1
ответ : - √6 ; √6 ; - 1 ; 1 ; - 2 ; 2
1) (x² + x + 1)(x² + x + 2) = 12
Сделаем замену :
x² + x + 1 = m , тогда x² + x + 2 = m + 1
m(m + 1) = 12
m² + m - 12 = 0
D = 1² - 4 * (- 12) = 1 + 48 = 49 = 7²
1) x² + x + 1 = - 4
x² + x + 5 = 0
D = 1² - 4 * 5 = 1 - 20 = - 19 < 0 - решений нет
2) x² + x + 1 = 3
x² + x - 2 = 0
По теореме Виета :
x₁ = - 2 x₂ = 1
ответ : - 2 ; 1
2)
3(x² + 5x + 1)² + 2x² + 10x = 3
3(x² + 5x + 1)² + 2(x² + 5x) = 3
Сделаем замену :
x² + 5x + 1 = m , тогда x² + 5x = m - 1
3m² + 2(m - 1) = 3
3m² + 2m - 2 - 3 = 0
3m² + 2m - 5 = 0
D = 2² - 4 * 3 * (- 5) = 4 + 60 = 64 = 8²
1)x²+ 5x + 1 = - 5/3
x² + 5x + 8/3 = 0
3x² + 15x + 8 = 0
D = 15² - 4 * 3 * 8 = 225 - 96 = 129
2)x²+ 5x + 1 = 1
x² + 5x = 0
x(x + 5) = 0
x₃ = 0 x₄ = - 5
3)
(x⁴ - 5x²)² - 2(x⁴ - 5x²) = 24
Сделаем замену :
x⁴ - 5x² = m
m² - 2m - 24 = 0
По теореме Виета :
m₁ = 6 m₂ = - 4
1) x⁴ - 5x² = 6
x⁴ - 5x² - 6 = 0
x² = 6
x₁ = - √6 x₂ = √6
x² = - 1 - решений нет
2) x⁴ - 5x² = - 4
x⁴ - 5x² + 4 = 0
x² = 4
x₃ = - 2 x₄ = 2
x² = 1
x₅ = - 1 x₆ = 1
ответ : - √6 ; √6 ; - 1 ; 1 ; - 2 ; 2