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Thread: Polynomials

  1. #1
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    Polynomials

    If $\displaystyle \alpha$ and $\displaystyle \beta$ are roots of the equation $\displaystyle x^2+mx+n=0$, find the roots of $\displaystyle nx^2+(2n-m^2)x+n=0$ in terms of $\displaystyle \alpha$ and $\displaystyle \beta$
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    Quote Originally Posted by nerdzor View Post
    If $\displaystyle \alpha$ and $\displaystyle \beta$ are roots of the equation $\displaystyle x^2+mx+n=0$, find the roots of $\displaystyle nx^2+(2n-m^2)x+n=0$ in terms of $\displaystyle \alpha$ and $\displaystyle \beta$
    From the first equation you can get n and m in terms of $\displaystyle \alpha$ and $\displaystyle \beta$:

    $\displaystyle \alpha \, \beta = n$.
    $\displaystyle \alpha + \beta = -m$.

    Solve the second equation using the quadratic formula. Then replace n and m in terms of $\displaystyle \alpha$ and $\displaystyle \beta$.
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    Help?

    Quote Originally Posted by mr fantastic View Post
    Solve the second equation using the quadratic formula. Then replace n and m in terms of $\displaystyle \alpha$ and $\displaystyle \beta$.
    How do I do the quadratic formula with the 2nd equation? It gets really messy and I can't get it. Help?
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    Quote Originally Posted by nerdzor View Post
    How do I do the quadratic formula with the 2nd equation? It gets really messy and I can't get it. Help?
    Sorry, but messy in maths is like digging a hole with a spade. It can be done but it takes effort. Your effort, not mine.

    If you're genuinely stuck (as in you have absolutley no idea what to do) I or someone else will give additional assistance (but not with the manual labour).
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    Quote Originally Posted by mr fantastic View Post
    Sorry, but messy in maths is like digging a hole with a spade. It can be done but it takes effort. Your effort, not mine.

    If you're genuinely stuck (as in you have absolutley no idea what to do) I or someone else will give additional assistance (but not with the manual labour).
    But, instead of a spade you might try using a bobcat by noting that if $\displaystyle \gamma$ and $\displaystyle \delta$ are roots of the second equation, then

    $\displaystyle \gamma \, \delta = 1$
    $\displaystyle \gamma + \delta = \frac{m^2}{n} - 2$

    where I've obviously divided the second equation through by n first .....
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