POTW #372: Solve $4x^6-6x^2+2\sqrt{2}=0$ without a calculator - June 25th, 2019

[anemone]

Here is this week's POTW:

-----

Solve the equation $4x^6-6x^2+2\sqrt{2}=0$ without the help of a calculator.

-----

Remember to read the https://mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to https://mathhelpboards.com/forms.php?do=form&fid=2!

 

[anemone]

Thanks to the following members for their correct solution!(Cool)

1. Olinguito
2. castor28
3. MegaMoh
4. Greg
5. kaliprasad

Solution from Olinguito:

Spoiler

Let $y=\dfrac1{x^2}$; then
$$4x^6-6x^2+2\sqrt2\ =\ 0$$
$\implies\ \dfrac4{y^3}-\dfrac6y+2\sqrt2\ =\ 0$

$\implies\ 2\sqrt2y^3-6y^2+4\ =\ 0$

$\implies\ z^3-3z^2+4\ =\ 0$ where $z=\sqrt2y$

$\implies\ (z-2)^2(z+1)\ =\ 0$

$\implies\ z\ =\ -1,\,2$

$\implies\ y\ =\ \dfrac z{\sqrt2}\ =\ -\dfrac1{\sqrt2},\,\sqrt2$

$\implies\ x^2\ =\ \dfrac1y\ =\ -\sqrt2,\,\dfrac1{\sqrt2}$

$\implies\ \boxed{x\ =\ \pm i\sqrt[4]2,\,\pm\dfrac1{\sqrt[4]2}}$.

(If only real solutions are required, then $\boxed{x\ =\ \pm\dfrac1{\sqrt[4]2}}$.)