[tex] x^{6}-9 x^{3}+ 8 = 0<=>x^{6}-x^{3}-8 x^{3}+ 8 = 0[/tex]
[tex] x^{3}(x^{3}-1)-8(x^{3}-1)=0 <=>(x^{3}-1)(x^{3}-8)=0[/tex]
Avem de rezolvat ecuatiile:
[tex] x^{3} -1=0[/tex]
[tex] x^{3}-8=0 [/tex]
Rezolvam ecuatia
[tex] x^{3} -1=0[/tex]
Observam ca x = 1 este una din cele 3 solutii ale ecuatiei.
Ecuatia este echivalenta cu:
x³ - x² + x² - x + x -1 = 0
Dam factor comun:
x²(x - 1) + x(x - 1) + 1(x - 1) = 0
(x - 1)(x² + x + 1) = 0
[tex] x_{1} = 1[/tex]
[tex] x_{23}= \frac{-1± \sqrt{1-4} }{2}=\frac{-1± \sqrt{-3}}{2} [/tex]
[tex] x_{2} = \frac{-1+i \sqrt{3} }{2}[/tex]
[tex] x_{3} = \frac{-1-i \sqrt{3} }{2}[/tex]
Rezolvam ecuatia
[tex] x^{3}-8=0 [/tex]
Observam ca x = 2 este una din cele 3 solutii ale ecuatiei.
Ecuatia este echivalenta cu:
x³ - 2x² + 2x² - 4x + 4x - 8 = 0
Dam factor comun:
x²(x - 2) + 2x(x - 2) + 4(x - 1) = 0
(x - 2)(x² + 2x + 4) = 0
[tex] x_{4} = 2[/tex]
[tex] x_{56}= \frac{-2± \sqrt{4-16} }{2}=\frac{-2± \sqrt{-12}}{2} [/tex]
=\frac{-2± 2\sqrt{-3}}{2} [/tex] =\frac{-1± 1\sqrt{-3}}{1} [/tex]
[tex] x_{5}=-1+i \sqrt{3} [/tex]
[tex] x_{6}=-1-i \sqrt{3} [/tex]
