Question

Rozwiąż równania;
x^{3}- 2x^{2} -5x+6=0

x^{6} - 7x^{3}-8=0

Answer 1

Zad 1
x³ - 2x² - 5x + 6 = 0
Trochę przekształcamy to równania
x³ - x² - x² + x - 6x + 6 = 0
x² (x - 1) - x (x - 1) - 6 (x - 1) = 0
(x - 1) (x² - x - 6) = 0
x - 1 = 0  ⇒ x = 1
lub
x² - x - 6 = 0
Δ = (-1)² - 4×1×(-6) = 1 + 24 = 25
√Δ = 5
x = (1 - 5) / 2×1 = -4 / 2 = -2
x = (1 + 5) / 2×1 = 6 / 2 = 3
Odpowiedż:
x = 1
x = -2
x = 3
Zad 2
x⁶ - 7x³ - 8 = 0
Podstawiamy za x³ np. t
t² - 7t - 8 = 0
Δ = (-7)² - 4×1×(-8) = 49 + 32 = 81
√Δ = 9
t₁ = (7 - 9) / 2 = -2/2 = -1
t₂ = (7 + 9) / 2 = 16 / 2 = 8
Wracamy do iksa
x³  = t₁ = -1
x = ∛(-1)
x = -1
x³ = t₂ = 8
x = ∛8
x = 2
Odpowiedź
x = -1
x = 2