Question

obtenha um numero real cujo quadrado,adicionando ao inverso do seu quadrado,resute em $\frac{5}{2}$

Answer 1

Olá, Anne.

$x^2+\frac1{x^2}=\frac52\Rightarrow\frac{2x^4+2}{2x^2}=\frac{5x^2}{2x^2}\Rightarrow2x^4-5x^2+2=0\\\\ \text{Fazendo a mudan\c{c}a de vari\'avel }y=x^2,\text{ temos:}\\\\ 2y^2-5y+2=0\Rightarrow y=\frac{5\pm\sqrt{25-16}}{4}\Rightarrow y_1=2\text{ e }y_2=\frac12\\\\ \text{Voltando \`a mudan\c{c}a de vari\'avel, temos:}\\\\ y_1=2=x_1^2\Rightarrow \boxed{x_1=\pm\sqrt2}\\\\ y_2=\frac12=x_2^2\Rightarrow \boxed{x_2=\pm\frac{\sqrt2}2}\\\\\\ \therefore\boxed{S=\{\pm\sqrt2;\pm\frac{\sqrt2}{2}\}}$