Question

As raízes da equação 3$x^{2}$+7x-18=0 são α e β. qual o valor de α²β+αβ²-α-β

Answer 1

3x² + 7x - 18 = 0
a = 3 / b = 7 / c = -18

∆ = b² - 4ac
∆ =7² - 4(3)(-18)
∆ = 49 + 216
∆ = 265
√∆ = √265

α = (-b + √∆)/2a
α = (-7 + √265)/6

β = (-b - √∆)/2a
β = (-7 - √265)/6

Logo:

α²β + αβ² - α - β

[(-7 + √265)/6]².[(-7 - √265)/6] + [(-7 + √265)/6].[(-7 - √265)/6]² - [(-7 + √265)/6] - [(-7 - √265)/6]

[(314 - 14√265)/36].[(-7 - √265)/6] + [(-7 + √265)/6].[(314 + 14√265)/36] - [(-7 + √265 + 7 - √265)/6]

[(-2198 - 314√265 + 98√265 + 3710)/216] + [(-2198 - 98√265 + 314√265 + 3710)/216] - 0

[(-2198 - 314√265 + 98√265 + 3710 - 2198 - 98√265 + 314√265 + 3710)/216)

[(-2198 - 2198 + 3710 + 3710)/216]

[3024/216] = 14

Resposta: 14.