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Thread: find all real solutions to an equation

  1. #1
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    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?
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  2. #2
    No one in Particular VonNemo19's Avatar
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    Detroit, MI
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    Quote Originally Posted by absvalue View Post
    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?
    They look good to me.
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