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Thread: solving equations w/ radicals and exponents problem, stuck

  1. #1
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    solving equations w/ radicals and exponents problem, stuck

    -6 - 2x = 4 times sqrt of x + 6

    this was a part of a larger problem but i see this is where it screws up, the answer is -5 as the solution set

    what ive been doin w/ problems before this one was divide that 4 in both places and it always fit nicely but not in this problem, iam really frustrated can someone gimmie a hint? thanks a bunch
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  2. #2
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    Re: solving equations w/ radicals and exponents problem, stuck

    heres the orginal problem

    sqrt of 4 - x - sqrt of x + 6 = 2
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  3. #3
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    Re: solving equations w/ radicals and exponents problem, stuck

    Quote Originally Posted by mathmathmathmathmathmathm View Post
    -6 - 2x = 4 times sqrt of x + 6

    this was a part of a larger problem but i see this is where it screws up, the answer is -5 as the solution set

    what ive been doin w/ problems before this one was divide that 4 in both places and it always fit nicely but not in this problem, iam really frustrated can someone gimmie a hint? thanks a bunch
    You can square both sides:

    $\displaystyle (-6-2x)^2 = (4\sqrt{x+6})^2 \ \Leftrightarrow \ 4x^2+24x+36 = 16(x+6)$

    Solve for x and check for extraneous solutions using the original equation. Note that due to the domain $\displaystyle x \geq -6$

    EDIT: Your equation in the OP is fine but there is an additional domain restriction $\displaystyle -6 \leq x \leq 4$

    I've put the working to show this in a spoiler to keep the post more coherent.


    Spoiler:
    Quote Originally Posted by mathmathmathmathmathmathm View Post
    heres the orginal problem

    $\displaystyle \sqrt{4 - x} - \sqrt{x + 6} = 2$
    $\displaystyle \sqrt{4-x} = 2 + \sqrt{x+6}$

    $\displaystyle 4-x = 4 + 4\sqrt{x+6} + x+6 = 10 + x + 4\sqrt{x+6}$

    If you tidy that up a bit: $\displaystyle -6-2x = 2\sqrt{x+6} $
    Last edited by e^(i*pi); Aug 5th 2011 at 12:16 PM. Reason: quoting OP+putting the original question into a spoiler
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  4. #4
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    Re: solving equations w/ radicals and exponents problem, stuck

    Quote Originally Posted by mathmathmathmathmathmathm View Post
    -6 - 2x = 4 times sqrt of x + 6

    this was a part of a larger problem but i see this is where it screws up, the answer is -5 as the solution set

    what ive been doin w/ problems before this one was divide that 4 in both places and it always fit nicely but not in this problem, iam really frustrated can someone gimmie a hint? thanks a bunch
    Is your equation $\displaystyle \displaystyle -6-2x = 4\sqrt{x} + 6$ or $\displaystyle \displaystyle -6-2x = 4\sqrt{x+6}$?
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  5. #5
    Junior Member BERMES39's Avatar
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    Re: solving equations w/ radicals and exponents problem, stuck

    Try this,
    Attached Thumbnails Attached Thumbnails solving equations w/ radicals and exponents problem, stuck-scan0007.jpg  
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  6. #6
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    Re: solving equations w/ radicals and exponents problem, stuck

    To avoid producing extraneous solutions I like the following method.

    $\displaystyle -6-2x = 4\sqrt{x+6}$ , but first, divide both sides by 2 & get rid of the terms on the left side.

    Starting with: $\displaystyle 0 = x+2\sqrt{x+6}+3$

    now add & subtract 6 at strategic locations: $\displaystyle 0 =x+6+2\sqrt{x+6}+3-6$

    $\displaystyle 0 = (\sqrt{x+6})^2+2\sqrt{x+6}-3$

    Now factor: $\displaystyle 0 = (\sqrt{x+6}+3)(\sqrt{x+6}-1)$

    Since $\displaystyle \sqrt{x+6}\ne-3$, the only solution can be $\displaystyle \sqrt{x+6}=1\,.$

    This gives x=-5 .
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  7. #7
    Junior Member mrmohamed's Avatar
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    Re: solving equations w/ radicals and exponents problem, stuck

    Quote Originally Posted by mathmathmathmathmathmathm View Post
    -6 - 2x = 4 times sqrt of x + 6

    this was a part of a larger problem but i see this is where it screws up, the answer is -5 as the solution set

    what ive been doin w/ problems before this one was divide that 4 in both places and it always fit nicely but not in this problem, iam really frustrated can someone gimmie a hint? Thanks a bunch
    hi all
    Attached Thumbnails Attached Thumbnails solving equations w/ radicals and exponents problem, stuck-codecogseqn-26-.gif  
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