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B Solution set of a square root

  1. May 8, 2017 #1
    When we find solution set of an equation inside a square root why we should assume that inside of square root should be equal to or greater than zero? For example ##\sqrt{5x-4}##.

    How can I use here equal to or greater than zero symbol?

    Thank you.
     
  2. jcsd
  3. May 8, 2017 #2

    BvU

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    As long as you are restricting yourself to real numbers, the expression under the square root sign has to be non-negative, so you can square if you want, but then you need to add & ##5x -4 \ge 0##
     
  4. May 8, 2017 #3
    For only real number, I cannot see that reason which restricting inside a square root being ##\ge 0##? I think it is not easy to see this. Yes if I take the square of both sides it is ##5x-4\ge 0##. This is an easy step but can it give any explanation?

    Thank you.
     
  5. May 8, 2017 #4

    BvU

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    There is no real number ##x## for which ##x^2 < 0 ##
     
  6. May 8, 2017 #5
    But there can be ## real\ numbers ## which can make ##5x-4## negative so when putting ##5x-4## inside a square root why should these values be neglected? I cannot understand this part easily.

    Thank you.
     
  7. May 8, 2017 #6

    DrClaude

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    Because sometimes a complex solution is not acceptable. If you have ##t = \sqrt{5x-4}##, where ##t## is time, then only real solutions are physically acceptable.
     
  8. May 8, 2017 #7
    Without thinking both complex numbers and meaningful physical situation, i.e from respect of pure real numbers, what is the reason?

    Thank you.
     
  9. May 8, 2017 #8

    DrClaude

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    If you restrict the codomain of a function to the real numbers, then that may restrict its domain.
     
  10. May 8, 2017 #9

    Mark44

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    Because for ##\sqrt{5x - 4}## to be real, ##5x - 4## must be greater than or equal to 0. Otherwise (if ##5x - 4 < 0##), you'll be taking the square root of a negative number.
     
  11. May 8, 2017 #10
    Yes this explains it but requires more thinking. Would you please check if this explanation expanding it correct?
    "If ##5x-4## negative real number and if we put it into square root ##\sqrt{5x-4}## cannot a real number because when we take square of ##\sqrt{5x-4}^2## this will be ##5x-4##, a negative real number, which is imposibble.

    Thank you.
     
    Last edited: May 8, 2017
  12. May 8, 2017 #11

    FactChecker

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    You should not assume that 5x-4 is ≥0. You should state that your answers are only valid for 5x-4 ≥0. That is significantly different.
    You can allow square roots of negative numbers if you are willing to talk about imaginary and complex numbers.

    PS. When you are editing a post, the ≥ symbol (and many others) is available by clicking the Σ symbol at the top right.
     
  13. May 8, 2017 #12
    But I regard only reel numbers, do not assume complex numbers.

    Thank you.
     
  14. May 8, 2017 #13

    FactChecker

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    I understand. Just a little "word-smithing". I don't like using the word "assume" when that may not be true. It's better to state a restriction that is required to make something valid than to assume it.
    Like: ##\sqrt{5x-4} = etc... ; x ≥ 4/5 ##
    Then you should also keep track of those restrictions when you give your final answer.
     
  15. May 8, 2017 #14
    But does not #10 explains corrrectly why this restriction should be?

    Thank you.
     
  16. May 8, 2017 #15

    FactChecker

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    Yes, it does. As long as you keep in mind that it is a restriction that you should keep track of, rather than an assumption that you don't need to worry about.
     
  17. May 8, 2017 #16

    FactChecker

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    Added to post #11 probably after you read it:
    When you are editing a post, the ≥ symbol (and many others) is available by clicking the Σ symbol at the top right.
     
  18. May 8, 2017 #17

    Mark44

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    You're making it much more complicated that it needs to be.
    From post #1,
    The usual square root function, the one that produces a real number, is defined only for expressions whose values are ≥ 0. This means that ##5x - 4 \ge 0## must be true, or equivalently, ##x \ge \frac 4 5##. In this case, if 5x - 4 < 0, its square root will not be a real number.​
    That's all you really need to say.
     
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