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When is following equation true?

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  1. Aug 27, 2015 #1

    Rectifier

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    1. The problem

    When is the following equation true
    ## \sqrt{c^2+14c+49} = c + 7##

    a) for all real c
    b) for ## c \geq -7 ##
    c) for ## c < -7##
    d) c > 0
    e) c < 0

    The attempt 1
    I know that the root of ## c^2+14c+49 = 0 ## is ## c = -7 ## and that this sqr-root is only defined for positive numbers. Thus the equation is true only when the stuff below the root is positive. But that stuff is always positive....

    The attempt 2
    ## \sqrt{c^2+14c+49} = c + 7 \\ \sqrt{(c+7)^2} = c + 7 \\ c+7 = c + 7 \\ ##
    Thus this equation is true for all real c:s. But somehow this is wrong.
     
  2. jcsd
  3. Aug 27, 2015 #2

    andrewkirk

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    Look at your last step in Attempt 2. Are you sure that it's always the case that ##\sqrt{x^2}=x##, given that the convention is that the positive square root is always implied by the square root sign? What about if x=-1? What happens if you start with -1, square it and then take the square root (which is by convention positive). Do you end up with the number you started with?
     
    Last edited: Aug 27, 2015
  4. Aug 27, 2015 #3

    Rectifier

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    The answer in my book :)
     
  5. Aug 27, 2015 #4

    DrClaude

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    The problem is in that last step, because ##\sqrt{x^2} \neq x##. Can you see why?

    What about trying it yourself?
     
  6. Aug 27, 2015 #5

    andrewkirk

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    Yeah, sorry, my first answer was too quick. I hope my redraft makes more sense.
     
  7. Aug 27, 2015 #6

    Rectifier

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    Yeah. Because negative x:es give different results.

    How can I implement that in my problem?
     
  8. Aug 27, 2015 #7

    DrClaude

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    You have to use absolute values.
     
  9. Aug 27, 2015 #8

    Rectifier

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    So basically |c + 7| = c + 7
     
  10. Aug 27, 2015 #9

    DrClaude

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    Yes. You should be able to convert that to a condition on ##c##.
     
  11. Aug 31, 2015 #10

    Mark44

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    To be clear, |c + 7| is not equal to c + 7, as when, for example, c = -8. I believe that @DrClaude is in agreement with this, but the casual reader might misinterpret his comment.

    ##\sqrt{(c + 7)^2} \neq c + 7##
    but
    ##\sqrt{(c + 7)^2} = |c + 7|##
     
  12. Aug 31, 2015 #11

    DrClaude

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    Of course I agree :smile:

    My point is that the question starts with:

    When is the following equation true
    ##\sqrt{c^2+14c+49} = c + 7##

    which is simplified to:

    When is the following equation true
    ##| c+ 7| = c + 7##

    from which it is easy to get a condition on ##c## for the original equation to be true.
     
  13. Sep 3, 2015 #12

    HallsofIvy

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    Use the definition of absolute value: [itex]|a|= a[/itex] if [itex]a\ge 0[/itex], [itex]|a|= -a[/itex] if [itex]a< 0[/itex].
     
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