Question

DETERMINE A SOLUCAO DA EQUACÃO X2+x+1=0
NO CONJUNTO DOS NUMEROS COMPLEXO.​

Answer 1

Explicação passo-a-passo:

$\sf x^2 + x + 1 = 0~~\Rightarrow~~\boxed{\sf x \mathbf{\in} \mathbb{C}}$ (x pertence ao conjunto dos complexos)

$\sf \Delta = b^2 - 4ac$

$\sf \Delta = 1^2 - 4\cdot1\cdot1$

$\sf \Delta = 1 - 4$

$\sf \Delta = - 3$

$\sf \Delta = (-1)(3)$

$\sf \Delta = (i^2)(3)$

$\sf x = \dfrac{- b \pm \sqrt{\Delta}}{2a}$

$\sf x = \dfrac{- 1 \pm \sqrt{(i^2)(3)}}{2\cdot1}$

$\sf x = \dfrac{- 1 \pm \sqrt{i^2} \cdot \sqrt{3}}{2}$

$\sf x = \dfrac{- 1 \pm i \cdot \sqrt{3}}{2}$

$\sf x = \dfrac{- 1 \pm i\sqrt{3}}{2}$

$•~~\sf x' = \dfrac{- 1 + i\sqrt{3}}{2}$

$•~~\sf x" = \dfrac{- 1 - i\sqrt{3}}{2}$

$\boxed{\sf{S = \left\{\dfrac{- 1 - i\sqrt{3}}{2}~~;~~\dfrac{- 1 + i\sqrt{3}}{2}\right\}}}$