find all real solutions to an equation

I'm asked to find all real solutions to the equation...

$\displaystyle \sqrt{2}x^{2} - 1 = 0$

Here is what I've done...

$\displaystyle \sqrt{2}x^{2} = 1$

$\displaystyle x^{2} = \frac{1}{\sqrt{2}}$

$\displaystyle x^{2} = \frac{\sqrt{2}}{2}$

$\displaystyle x = \stackrel{+}{-}\sqrt{\frac{\sqrt{2}}{2}}$

$\displaystyle x = \stackrel{+}{-}\frac{\sqrt[4]{2}}{\sqrt{2}}$

$\displaystyle x = \stackrel{+}{-}\frac{\sqrt[4]{2} \times \sqrt{2}}{2}$

$\displaystyle x = \stackrel{+}{-}\frac{\sqrt[4]{2^{3}}}{2}$

$\displaystyle x = \stackrel{+}{-}\frac{\sqrt[4]{8}}{2}$

I'm not feeling confident about this, though. Are these steps correct?