Question

Uma das raízes da equação x² + 32 = 0 é:
a) i√2
b) √32
c )2i√2
d) 3i√2
e) 4i√2

Answer 1

$\green{\boxed{\mathbf{LETRA~~E)}}}$

Explicação passo-a-passo:

$\sf{x^2 + 32 = 0}$

$\sf{x^2 = - 32}$

$\sf{x = \sqrt{- 32}}$

$\Rightarrow~\sf{x \mathbf{\notin} \mathbb{R}~~;~~\Rightarrow~x \mathbf{\in} \mathbb{C}}$

$\sf{x = \sqrt{(-1)\cdot(32)}}$

$\sf{x = \sqrt{-1} \cdot \sqrt{32}}$

$\sf{x = \sqrt{i^2} \cdot \sqrt{32}}$

$\sf{x = \backslash \!\!\! \sqrt{i ^\backslash\!\!\!^2} \cdot \sqrt{32}}$

$\sf{x = i \cdot \sqrt{32}}$

M.M.C (32)

$\begin{array}{r|l} 32&2 \\ 16&2 \\ 8&2 \\ 4&2 \\ 2&2 \\ 1 \end{array}~~\Rightarrow~~2^2 \cdot 2^2 \cdot 2$

$\sf{x = i \cdot \sqrt{2^\backslash \!\!\!^2\cdot 2^\backslash \!\!\!^2 \cdot 2}}$

$\sf{x = i \cdot 2\cdot2\sqrt{2}}$

$\sf{x = i \cdot 4\sqrt{2}}$

$\sf{x = \pm~~4i\sqrt{2}}$

$\sf{S = \left\{- 4i\sqrt{2}~~;~~4i\sqrt{2}\right\}}$

Sendo uma das raízes , de acordo com as alternativas, apenas possui a raiz positiva:

$\green{\boxed{\mathbf{LETRA~~E)}}}$