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- Feb 14, 2012

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

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- Thread starter
- Admin
- #1

- Feb 14, 2012

- 3,894

-----

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

-----

Remember to read the POTW submission guidelines to find out how to submit your answers!

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- Feb 14, 2012

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1. Olinguito

2. castor28

3. MegaMoh

4. Greg

5. kaliprasad

Solution from Olinguito :

$$4x^6-6x^2+2\sqrt2\ =\ 0$$

$\implies\ \dfrac4{y^3}-\dfrac6y+2\sqrt2\ =\ 0$

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

$\implies\ z^3-3z^2+4\ =\ 0$ 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 $\boxed{x\ =\ \pm\dfrac1{\sqrt[4]2}}$.)

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