Question

Determine a soluçao da equação x^4-65x^2+64>0

Answer 1

$\mathsf{Fixemos~~x^4=y^2~~e~~x^2=y}\\\\\\\mathsf{x^4-65x^2+64\ \textgreater \ 0}\\\mathsf{\hspace{6}\searrow_{y^2}\hspace{14}\searrow_{y}}\\\\\mathsf{y^2-65y+64\ \textgreater \ 0}\\\\\textsf{Calculemos ent\~ao suas ra\'izes~(farei~por~completamento~de~quadrados):}$

$\mathsf{y^2-65y+64\ \textgreater \ 0}\\\\\mathsf{y^2-65y\ \textgreater \ -64}\\\\\mathsf{y^2-65y+(\frac{65}{2})^2\ \textgreater \ -64+(\frac{65}{2})^2}\\\\\mathsf{(y-\frac{65}{2})^2\ \textgreater \ -64+\frac{4225}{4}}\\\\\mathsf{(y-\frac{65}{2})^2\ \textgreater \ -\frac{256}{4}+\frac{4225}{4}}\\\\\mathsf{(y-\frac{65}{2})^2\ \textgreater \ \frac{3969}{4}}\\\\\mathsf{\sqrt{(y-\frac{65}{2})^2}\ \textgreater \ \sqrt{\frac{3969}{4}}}\\\\\mathsf{|y-\frac{65}{2}|\ \textgreater \ \frac{63}{2}~\Longleftrightarrow~y-\frac{65}{2}\ \textless \ -\frac{63}{2}~~~~\vee~~~~y-\frac{65}{2}\ \textgreater \ \frac{63}{2}}$


$\begin{Bmatrix}\mathsf{y\ \textless \ 1}\\\\\mathsf{ou}\\\\\mathsf{y\ \textgreater \ 64}\end.\\\\\\\\\mathsf{Como~x^2=y,~temos:}\\\\\\\\\begin{Bmatrix}\mathsf{x^2\ \textless \ 1~\rightarrow~|x|\ \textless \ 1~\rightarrow~-1\ \textless \ x\ \textless \ 1\hspace{26}}\\\\\mathsf{ou}\\\\\mathsf{x^2\ \textgreater \ 64~\rightarrow~|x|\ \textgreater \ 8~\rightarrow~x\ \textless \ -8~~ou~~x\ \textgreater \ 8}\end.\\\\\\\\\fbox{ \mathsf{S=\begin{Bmatrix}\mathsf{x\in\mathbb{R}|~x\ \textless \ -8~~~ou~~~-1\ \textless \ x\ \textless \ 1~~~ou~~~x\ \textgreater \ 8}\end{Bmatrix}} }~~\mathsf{(resposta)}$

$\mathsf{Reta~~\mathbb{R}:x^4-65x^2+64\ \textgreater \ 0}\\\\\\\mathsf{\underline{~++++~}\underset{\hspace{-6}-8}\circ\underline{~----~}\underset{\hspace{-6}-1}\circ\underline{~++++~}\underset{1}\circ\underline{~----~}\underset{8}\circ}\underline{~++++~}_\blacktriangleright\underset{_\mathsf{x}}$

Gráfico (anexo):