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Math Help - Rational Prob.

  1. #1
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    Rational Prob.

    If (3+x)(3-x) is rational, then x is rational.

    how would I claim that (3+x)(3-x) = p/q

    Would I put (3+x)(3-x)/1=p/q -> q(3+x)(3-x)/p?
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  2. #2
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    Re: Rational Prob.

    Quote Originally Posted by xmathlover View Post
    If (3+x)(3-x) is rational, then x is rational.

    how would I claim that (3+x)(3-x) = p/q

    Would I put (3+x)(3-x)/1=p/q -> q(3+x)(3-x)/p?
    I don't think this is true, take (3-x)(3+x)=1 for instance. We have x^2=8 \implies x =2\sqrt{2}

    The converse, however, is true. If x=\frac{p}{q} then (3+x)(3-x)=9-x^2=9-\frac{p^2}{q^2}=\frac{9q^2-p^2}{q^2}
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  3. #3
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    Re: Rational Prob.

    I think you meant to write (3+x)/(3-x). Then this is true.

    Assume that (3 + x)/(3 - x) is rational. Then (3 + x) / (3 - x) = p/q for some integers p and q where q =/= 0. Then cross multiply.

    Thus, q(3 + x) = p(3 - x).
    Then 3q + qx = 3p - px
    Then px + qx = 3p - 3q
    Then x(p + q) = 3(p - q)
    Then x = 3(p - q)/(p + q)

    Since p and q are integers, p - q and p + q are integers. Then 3(p - q) and p + q are integers. Then 3(p - q)/(p + q) is rational. Thus, x is rational.
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