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Math Help - a rational number?

  1. #1
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    a rational number?

    Hello. I'd like to ask if there is any possibility that
    sqrt(7) - sqrt(x)
    can give a rational number?
    Please, give the proof.
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  2. #2
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    Quote Originally Posted by olaolaola View Post
    Hello. I'd like to ask if there is any possibility that
    sqrt(7) - sqrt(x)
    can give a rational number?
    Please, give the proof.
    Hint: 0 is rational
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  3. #3
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    yes, i meant except of 0?
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  4. #4
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    Quote Originally Posted by olaolaola View Post
    yes, i meant except of 0?
    Solve \sqrt{7}-\sqrt{x}=1 for x.
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  5. #5
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    Yes, I see. But in this equation x is not a rational number. And if in my equation: sqrt(7) - sqrt(x) , x is a rational number? Then can the result be somehow rational (and not 0)?
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  6. #6
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    Quote Originally Posted by olaolaola View Post
    Yes, I see. But in this equation x is not a rational number. And if in my equation: sqrt(7) - sqrt(x) , x is a rational number? Then can the result be somehow rational (and not 0)?
    You do understand this wastes people's time, if you don't specify the question clearly and we end up answering the wrong question?

    Let a,b\in\mathbb{Z},b\ne0

    \sqrt{7}-\sqrt{x}=\frac{a}{b}

    \sqrt{7}-\frac{a}{b}=\sqrt{x}

    x=(\sqrt{7}-\frac{a}{b})^2

    x = 7-\left(\frac{2a}{b}\right)\sqrt{7}+\frac{a^2}{b^2}

    With a little more work we see that the only way to make x rational is to let a = 0.
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  7. #7
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    Yes, I understand, next time I promise I will be more precise. But I'm very grateful, you helped me very much. Thank you!
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  8. #8
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    x=7
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