Question

X⁴-5X²-10X-6=0 resolver a equação sabendo que duas raízes são -1 e 3

Answer 1

Resposta:

$\textsf{Leia abaixo}$

Explicação passo a passo:

$\mathsf{x^4 - 5x^2 - 10x - 6 = 0}$

$\mathsf{x_1 = -1}$

$\mathsf{x_2 = 3}$

$\mathsf{(x - x_1)(x - x_2)\:.\:p(x) = x^4 - 5x^2 - 10x - 6}$

$\mathsf{(x + 1)(x - 3)\:.\:p(x) = x^4 - 5x^2 - 10x - 6}$

$\mathsf{(x^2 -2x - 3)\:.\:p(x) = x^4 - 5x^2 - 10x - 6}$

$\mathsf{p(x) = \left(\dfrac{x^4 - 5x^2 - 10x - 6}{x^2 -2x - 3}\right)}$

$\mathsf{p(x) = x^2 + 2x + 2}$

$\mathsf{\Delta = b^2 - 4.a.c}$

$\mathsf{\Delta = (2)^2 - 4.1.2}$

$\mathsf{\Delta = 4 - 8}$

$\mathsf{\Delta = -4}$

$\mathsf{x = \dfrac{-b \pm \sqrt{\Delta}}{2a} = \dfrac{-2 \pm \sqrt{-4}}{2} \rightarrow \begin{cases}\mathsf{x' = \dfrac{-2 + 2i}{2} = -1 + i}\\\\\mathsf{x'' = \dfrac{-2 - 2i}{2} = -1 - i}\end{cases}}$

$\boxed{\boxed{\mathsf{S = \{-1;3;(-1 + i);(-1 - i)\}}}}$