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

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

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

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

my approach a bit tedious, expecially the quadratic formula part, there might be a shortcut?

3. or maybe you can let the roots of $\displaystyle nx^{2}+(2n-m^2)x+n=0$ be a and b.

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

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

So $\displaystyle a=\frac{\alpha}{\beta}$ and $\displaystyle b=\frac{\beta}{\alpha}$

LOL