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Math Help - Quickie #2

  1. #1
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    Quickie #2


    Solve: . \sqrt[3]{6x + 28} - \sqrt[3]{6x - 28} \:=\:2


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  2. #2
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    This is some wacky thing, Soroban, or does it supposed to have a viable answer.

    If so, I got -6 and 6
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  3. #3
    Senior Member OReilly's Avatar
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    Quote Originally Posted by Soroban View Post

    Solve: . \sqrt[3]{6x + 28} - \sqrt[3]{6x - 28} \:=\:2


    \begin{array}{l}<br />
 \sqrt[3]{{6x + 28}} - \sqrt[3]{{6x - 28}} = 2 \\ <br />
 \left( {\sqrt[3]{{6x + 28}} - \sqrt[3]{{6x - 28}}} \right)^3  = 2^3  \\ <br />
 \end{array}<br />
    6x + 28 - 3\sqrt[3]{{6x + 28}}\sqrt[3]{{6x - 28}}(\sqrt[3]{{6x + 28}} - \sqrt[3]{{6x - 28}}) - 6x + 28 = 8
     - 3\sqrt[3]{{6x + 28}}\sqrt[3]{{6x - 28}}(\sqrt[3]{{6x + 28}} - \sqrt[3]{{6x - 28}}) =  - 48
     - 3\sqrt[3]{{6x + 28}}\sqrt[3]{{6x - 28}}(2) =  - 48
    \begin{array}{l}<br />
 3\sqrt[3]{{(6x + 28)(6x - 28)}} = 24 \\ <br />
 \sqrt[3]{{(6x + 28)(6x - 28)}} = 8 \\ <br />
 (6x + 28)(6x - 28) = 512 \\ <br />
 36x^2  - 784 = 512 \\ <br />
 36x^2  = 1296 \\ <br />
 x^2  = 36 \\ <br />
 x =  \pm 6 \\ <br />
 \end{array}<br />
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  4. #4
    Eater of Worlds
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    Soroban has a clever trick to solve this.
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  5. #5
    Senior Member OReilly's Avatar
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    Quote Originally Posted by ThePerfectHacker View Post
    I think the trick is that we need to check the solutions. Because those the necessary but not sufficient conditions.
    Thus,
    x\not = -6
    Why x=-6 isn't solution?

    \sqrt[3]{{6( - 6) + 28}} - \sqrt[3]{{6( - 6) - 28}} = \sqrt[3]{{ - 8}} - \sqrt[3]{{ - 64}} =  - 2 - ( - 4) = 2
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  6. #6
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    Hello, OReilly!

    Well done!

    The "Quickie" solution uses a clever theorem,
    . . but is not much shorter than your solution.

    Theorem: .If a + b + c .= .0, then: .a + b + c .= . 3abc **

    . . . . . . . . . _______ . . ______
    We have: .√6x + 28 - √6x - 28 - 8 .= .0
    -. . . . . . . . . . . . . . . . . . . w .
    . . . . . . . . . . . a . . . . . . . b . o . c

    The thereom gives us: . . . . . . . . ._________________
    . . (6x + 28) - (6x - 28) - 8 .= .3√(6x + 28)(6x - 28)(8)
    . . . . . . . . . . . . . . . . . . . . . . . . _________
    . . . . . . . . . . . . . . . . .48 .= .6√36x - 784

    Then: .36x - 784 .= .512 . . x = 36 . . x = 6

    ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

    ** . Proof

    We have: .a + b .= .-c

    Cube both sides: .(a + b) .= .(-c)

    Expand: .a + 3ab + 3ab + b .= .-c

    Then: .a + b + c .= .-3ab - 3ab

    . . . . . a + b + c .= .-3ab(a + b)


    Since a + b = -c, we have: .a + b + c .= .-3ab(-c)

    Therefore: . a + b + c .= .3abc

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