Question

Fie a = (1-√5)/2
a) Aratati ca a² - a - 1 = 0
b) Calculati (3a² - 3a - 4)²⁰²¹
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Answer 1

$a = \dfrac{1 - \sqrt{5} }{2}$

a)

$a^{2} = \bigg(\dfrac{1 - \sqrt{5} }{2} \bigg)^{2} = \dfrac{1 -2\sqrt{5} + 5}{4} = \dfrac{2(3 -\sqrt{5})}{4} = \dfrac{3 -\sqrt{5}}{2}$

$a^{2} - a - 1 = \dfrac{3 -\sqrt{5}}{2} - \dfrac{1 -\sqrt{5}}{2} - 1 = \dfrac{3 -\sqrt{5} - 1 + \sqrt{5} - 2}{2} = \dfrac{3 - 3}{2} =\bf 0$

b)

$3a^{2} - 3a - 4 = 3a^{2} - 3a - 3 - 1 = 3(a^{2} - a - 1) - 1 = 3 \cdot 0 - 1 = 0 - 1 = -1$

$(3a^{2} - 3a - 4)^{2021} = (-1)^{2021} =\bf -1$