Question

Resolva a equação do segundo grau;
2x² + 2x + 1 = 0, em complexos

Answer 1

Explicação passo-a-passo:

Resolver equação no conjunto dos complexos $\sf \left\{x \mathbf{\in} \mathbb{C}\right\}$:

$\sf 2x^2 + 2x + 1 = 0$

coeficientes:

a = 2 , b = 2 , c = 1

Resolução:

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

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

$\sf \Delta = 4 - 8$

$\sf \Delta = - 4$

$\sf \Delta = (-1)(4)$

$\sf \Delta = (i^2)(4)$

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

$\sf x = \dfrac{- 2 \pm \sqrt{(i^2)(4)}}{2 \cdot 2}$

$\sf x = \dfrac{- 2 \pm \sqrt{(i^2)} \cdot \sqrt{(4)}}{4}$

$\sf x = \dfrac{- 2 \pm i \cdot 2}{4}$

$\sf x = \dfrac{- 2 \pm 2i}{4}$

$\sf •~~x = \dfrac{- 2 + 2i}{4} = \dfrac{- 1 + i}{2}$

$\sf •~~x = \dfrac{- 2 - 2i}{4} = \dfrac{- 1 - i}{2}$

$\boxed{\sf S = \left\{\dfrac{- 1 + i}{2}~~;~~\dfrac{- 1 - i}{2}\right\}}</p><p>$