Question

A equação x^4 – 6x² + c = 0 admite quatro raízes reais distintas para quais valores?
A) –1< c < 9.
B) –9 < c < 9.
C) –3 < c < 3.
D) 0 < c < 3.
E) 0 < c < 9.

Answer 1

$\text x^4-6\text x^2+\text c=0\\\\ \text x^4-6\text x^2+9+\text c =9 \\\\ (\text x^2-3)^2=9-\text c \\\\ \text x^2-3=\pm\sqrt{9-\text c} \\\\ \underline{\text{Primeira restri{\c c}{\~a}o }}: \\\\ 9-\text c > 0 \to \boxed{\text c < 9} \\\\ \underline{\text{Continuando}}: \\\\ \text x^2 = 3\pm\sqrt{9-\text c} \\\\ \text x=\pm\sqrt{3\pm\sqrt{9-\text c}} \\\\ \text{Pegando uma das ra{\'i}zes para analisar}: \\\\\ \sqrt{3-\sqrt{9-\text c}} >0\\\\ 3-\sqrt{9-\text c}>0 \\\\ \sqrt{9-\text c}<3 \\\\ 9-\text c<9$

$-\text c<0 \to \boxed{\text c>0}$

Portanto :

$\huge\boxed{\ 0<\text c<9\ }\checkmark$

Letra E