Question

A soma dos cubos das raízes da equação x²- 3x - 5 = 0 é



a) 9

b) 18

c) 36

d) 72

e) 81

Answer 1

Resposta:

d) 72

Raízes

x1 ≈ 4.192582404

x2 ≈ 1.192582404

Answer 2

Resposta:

$Alternativa\ d.$

Explicação passo a passo: Primeiro a gente encontra as raizes, depois eleva elas ao cubo e somamos. Segue a solução.

$x^2-3x-5=0\\\\a=1,\ b=-3\ e\ c=-5\\\\x_{1,2}=\frac{-b\±\sqrt{b^2-4ac}}{2a}\\\\x_{1,2}=\frac{-(-3)\±\sqrt{(-3)^2-4(1)(-5)}}{2(1)}\\\\x_{1,2}=\frac{3\±\sqrt{9+20}}{2}\\\\x_{1,2}=\frac{3\±\sqrt{29}}{2}\\\\x_{1}=\frac{3+\sqrt{29}}{2}\ e\ x_{2}=\frac{3-\sqrt{29}}{2}\\\\\\\\Elevando\ ao\ cubo\ as\ duas\ raizes\ e\ somando:\\\\(\frac{3+\sqrt{29}}{2})^3+(\frac{3-\sqrt{29}}{2})^3$

$[(\frac{3+\sqrt{29}}{2})(\frac{3+\sqrt{29}}{2})(\frac{3+\sqrt{29}}{2})]+[(\frac{3-\sqrt{29}}{2})(\frac{3-\sqrt{29}}{2})(\frac{3-\sqrt{29}}{2})]\\\\(\frac{9+3\sqrt{29}+3\sqrt{29}+29}{4})(\frac{3+\sqrt{29}}{2})+(\frac{9-3\sqrt{29}-3\sqrt{29}+29}{4})((\frac{3-\sqrt{29}}{2})\\\\(\frac{38+6\sqrt{29}}{4})(\frac{3+\sqrt{29}}{2})+(\frac{38-6\sqrt{29}}{4})(\frac{3-\sqrt{29}}{2})$

$(\frac{19+3\sqrt{29}}{2})(\frac{3+\sqrt{29}}{2})+(\frac{19-3\sqrt{29}}{2})(\frac{3-\sqrt{29}}{2})\\\\(\frac{57+19\sqrt{29}+9\sqrt{29}+3(29)}{4})+(\frac{57-19\sqrt{29}-9\sqrt{29}+3(29)}{4})\\\\(\frac{57+87+28\sqrt{29}}{4})+(\frac{57+87-28\sqrt{29}}{4})$

$(\frac{144+28\sqrt{29}}{4})+(\frac{144-28\sqrt{29}}{4})\\\\36+7\sqrt{29}+36-7\sqrt{29}\\\\72+(7\sqrt{29}+36-7\sqrt{29})\\\\72+0\\\\72$