Answer:
[tex]$x^2=\frac{9}{4} \implies x=\pm \frac{3}{2} $[/tex]
[tex]$x^2=-2 \implies x=\pm\sqrt{-2} \implies x=\pm\sqrt{2i} $[/tex]
The solutions are
[tex]$\{-\frac{3}{2},\frac{3}{2}, -\sqrt{2i}, \sqrt{2i} \}$[/tex]
The Real roots are
[tex]$\{-\frac{3}{2},\frac{3}{2}\}$[/tex]
Step-by-step explanation:
[tex]$\frac{18}{x^4} +\frac{1}{x^2}=4$[/tex]
Multiply both sides by [tex]x^4[/tex]
[tex]$x^4\frac{18}{x^4} +x^4\frac{1}{x^2}=4\cdot x^4$[/tex]
[tex]$18+x^2=4x^4$[/tex]
[tex]4x^4-x^2-18=0[/tex]
Substitute [tex]x^2=t[/tex]
[tex]4t^2-t-18=0[/tex]
Solving the quadratic equation using the quadratic formula:
[tex]$t=\frac{-(-1)\pm \sqrt{(-1)^2-4\cdot 4(-18)}}{2\cdot 4}$[/tex]
[tex]$t=\frac{1\pm \sqrt{289}}{8}$[/tex]
[tex]$t=\frac{1\pm 17}{8}$[/tex]
[tex]$t_1=\frac{9}{4}$[/tex]
[tex]t_2=-2[/tex]
Now we have to solve for [tex]x^2=t[/tex]
[tex]$x^2=\frac{9}{4} \implies x=\pm \frac{3}{2} $[/tex]
[tex]$x^2=-2 \implies x=\pm\sqrt{-2} \implies x=\pm\sqrt{2i} $[/tex]
