Question

Quais valores de x satisfazem a seguinte equação:

Answer 1

Resposta:

Resposta A.

Explicação passo-a-passo:

$\frac{\left(x^2+1\right)}{4}+\frac{1}{x^2}=\frac{3}{2}\\\\\frac{x^2+1}{4}+\frac{1}{x^2}=\frac{3}{2}\\\\\mathrm{Encontrar\:o\:MMC\:de\:}4,\:x^2,\:2:\\\mathrm{Multiplicar\:pelo\:MMC=}4x^2\\\\\frac{x^2+1}{4}\cdot \:4x^2+\frac{1}{x^2}\cdot \:4x^2=\frac{3}{2}\cdot \:4x^2\\\\x^2\left(x^2+1\right)+4=6x^2\\\\\mathrm{Resolver\:}\:x^2\left(x^2+1\right)+4=6x^2\\\\x^4+x^2+4=6x^2\\x^4+x^2+4-6x^2=6x^2-6x^2\\x^4-5x^2+4=0\\\\\mathrm{Reescrever\:com\:}u=x^2\mathrm{\:e\:}u^2=x^4\\u^2-5u+4=0$

$\mathrm{Para\:}\quad a=1,\:b=-5,\:c=4:\quad u_{1,\:2}=\frac{-\left(-5\right)\pm \sqrt{\left(-5\right)^2-4\cdot \:1\cdot \:4}}{2\cdot \:1}\\u=4,\:u=1\\\\\mathrm{Sendo\:}u=x^2\mathrm{,\:resolver\:as\:anteriores\:para\:encontrar\:}x\\\\x^2=4\\x=\sqrt{4},\:x=-\sqrt{4}\\x=2,\:x=-2\\\\x^2=1\\x=\sqrt{1},\:x=-\sqrt{1}\\x=1,\:x=-1\\\\x=2,\:x=-2,\:x=1,\:x=-1$