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  1. #1
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    surds

    Express $\displaystyle \sqrt{66-24\sqrt{6}} $in the form of $\displaystyle p\sqrt{2}+q\sqrt{3}$

    I got $\displaystyle p=-3$ and $\displaystyle q=4$ or$\displaystyle p=3 $and $\displaystyle q=-4 $

    The answer is$\displaystyle p=-3$ and $\displaystyle q=4$ . My question is why is$\displaystyle p=3 $and $\displaystyle q=-4$ ommitted ??


    Thanks .
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  2. #2
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    Hello,
    Quote Originally Posted by thereddevils View Post
    Express $\displaystyle \sqrt{66-24\sqrt{6}} $in the form of $\displaystyle p\sqrt{2}+q\sqrt{3}$

    I got $\displaystyle p=-3$ and $\displaystyle q=4$ or$\displaystyle p=3 $and $\displaystyle q=-4 $

    The answer is$\displaystyle p=-3$ and $\displaystyle q=4$ . My question is why is$\displaystyle p=3 $and $\displaystyle q=-4$ ommitted ??


    Thanks .
    That's because $\displaystyle \sqrt{66-24\sqrt{6}}$ is positive (it's a square root.. if it was $\displaystyle -\sqrt{66-24\sqrt{6}}$, it would surely be negative)

    Now, have a thought on this...
    $\displaystyle \sqrt{2}<\sqrt{3}$
    Then
    $\displaystyle 3\sqrt{2}<3\sqrt{3}<4\sqrt{3}$

    So if p=3 and q=-4, would $\displaystyle p\sqrt{2}+q\sqrt{3}$ be positive ??
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  3. #3
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    Quote Originally Posted by thereddevils View Post
    Express $\displaystyle \sqrt{66-24\sqrt{6}} $in the form of $\displaystyle p\sqrt{2}+q\sqrt{3}$

    I got $\displaystyle p=-3$ and $\displaystyle q=4$ or$\displaystyle p=3 $and $\displaystyle q=-4 $

    The answer is$\displaystyle p=-3$ and $\displaystyle q=4$ . My question is why is$\displaystyle p=3 $and $\displaystyle q=-4$ ommitted ??


    Thanks .
    Hi

    You effectively get 2 possible solutions since $\displaystyle \left(3\sqrt{2}-4\sqrt{3}\right)^2 = \left(-3\sqrt{2}+4\sqrt{3}\right)^2 = 66-24\sqrt{6}$

    But $\displaystyle \sqrt{66-24\sqrt{6}} > 0$ and $\displaystyle -3\sqrt{2}+4\sqrt{3} > 0$ but $\displaystyle 3\sqrt{2}-4\sqrt{3} < 0$
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