# Thread: solving polynomials with roots

1. ## 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.

2. 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?

3. 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 $
b=\frac{\beta}{\alpha}
$

LOL