Объяснение:
А) 5x^3 - 3x^2 - 3x + 5 = 0
5x^3 + 5 - 3x^2 - 3x = 0
5(x^3 + 1) - 3x(x + 1) = 0
5(x + 1)(x^2 - x + 1) - 3x(x + 1) = 0
(x + 1)(5x^2 - 5x + 5 - 3x) = 0
(x + 1)(5x^2 - 8x + 5) = 0
x + 1 = 0 => x = -1
5x^2 - 8x + 5 = 0
D = 8^2 - 4 * 5 * 5 = 64 - 100 = -36
∅
ответ: x = -1
Б) (x + 1/x)^2 - 5(x + 1/x) + 6 = 0
t = x + 1/x
t^2 - 5t + 6 = 0
D = 5^2 - 4 * 1 * 6 = 25 - 24 = 1
t1 = (5 + 1) / 2 = 6/2 = 3
t2 = (5 - 1) / 2 = 4/2 = 2
x + 1/x = 3
x^2 - 3x + 1 = 0
D = 3^2 - 4 * 1 * 1 = 9 - 4 = 5
x1 = (3 - √5) / 2
x2 = (3 + √5) / 2
x + 1/x = 2
x^2 - 2x + 1 = 0
D = 2^2 - 4 * 1 * 1 = 4 - 4 = 0
x = 2 / 2 = 1
ответ: x1 = (3 - √5) / 2 ; x2 = 1 ; x3 = (3 + √5) / 2.
5x³ - 3x² - 3x + 5 = 0
5x³ +5 - 3x² - 3x = 5(x³ + 1) - 3x(x + 1) = 5(x + 1)(x² - x + 1) -3x(x + 1) = (x + 1)(5x² -5x + 5 - 3x) = (x + 1)(5x² - 8x + 5) = 0
x = -1
5x² - 8x + 5 = 0
D = 64 - 80 < 0
x ∈ ∅ при x ∈ R
ответ -1
(x + 1/x)² - 5(x + 1/x) + 6 = 0
x ≠ 0
x + 1/x = t
t² - 5t + 6 = 0
D = 25 - 24 = 1
t12 = (5 +- 1)/2 = 2 3
1. t = 2
x + 1/x = 2
(x² - 2x + 1)/x = 0
(x - 1)²/ x = 0
x = 1
2. t = 3
x + 1/x = 3
(x² - 3x + 1)/x = 0
D = 9 - 4 = 5
x12 = (3 +- √5)/2
ответ (3 +- √5)/2, 1
x⁴ - 5x³ + 8x² - 5x + 1 = 0
x ≠ 0
разделим на x²
1/x² + x² = 1/x² + 2*x²*1/x² + x² - 2*x²*1/x² = (x + 1/x)² - 2
x² - 5x + 8 - 5/x + 1/x² = x² + 1/x² - 5(x + 1/x) + 8 = (x + 1/x)² - 2 - 5(x + 1/x) + 8 = (x + 1/x)² - 5(x + 1/x) + 6 = 0
x + 1/x = t
t² - 5t + 6 = 0
это уравнение было номер 2
D = 25 - 24 = 1
t12 = (5 +- 1)/2 = 2 3
1. t = 2
x + 1/x = 2
(x² - 2x + 1)/x = 0
(x - 1)²/ x = 0
x = 1
2. t = 3
x + 1/x = 3
(x² - 3x + 1)/x = 0
D = 9 - 4 = 5
x12 = (3 +- √5)/2
ответ (3 +- √5)/2, 1