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Math Help - solving polynomials with roots

  1. #1
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    solving polynomials with roots

    Hi, I have a very simple problem which i don't seem to be able to get. If you could help out that would be great!

    If alpha and beta are roots of the equation x^2 + mx + n = 0, find the roots of nx^2 + (2n-m^2)x + n = 0 in terms of alpha and beta.

    Thanks.
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  2. #2
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    Let \alpha & \beta be the roots of x^{2}+mx+n=0,

    Then \alpha + \beta =-m and \alpha \beta = n
    So m=-(\alpha + \beta) and n=\alpha \beta

    Use the quadratic formula to solve x for the equation nx^{2}+(2n-m^2)x+n=0, then substitute your m and n and simplify the two roots.

    I got \frac{\alpha}{\beta} and \frac{\beta}{\alpha}

    my approach a bit tedious, expecially the quadratic formula part, there might be a shortcut?
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  3. #3
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    or maybe you can let the roots of nx^{2}+(2n-m^2)x+n=0 be a and b.

    Then a+b=\frac{m^{2}-2n}{n}=...= \frac{\alpha^{2}+\beta^{2}}{\alpha\beta}=\frac{\al  pha}{\beta}+\frac{\beta}{\alpha}

    ab=1=\frac{\alpha}{\beta}\times\frac{\beta}{\alpha  }

    So a=\frac{\alpha}{\beta} and <br />
b=\frac{\beta}{\alpha}<br />

    LOL
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