<s>[INST] $ax^2+bx+c=0$ has roots $x_1,x_2$. what are the roots of $cx^2+bx+a=0$. Given solution: Dividing the first equation by $x^2$ we get
$c(\frac{1}{x^2})+b(\frac{1}{x})+a=0$
so $(\frac{1}{x_1}),(\frac{1}{x_2})$ are the roots of $cx^2+bx+a=0$.{How?It is not obvious to me.}
The answers so far are proving retrospectively that the roots are indeed those given above.I would like to know how the relation of the roots is derived.
 [/INST] Another way to see it is to look at sum and product of roots. $x_1$ and $x_2$ are roots of $ax^2+bx+c=0$ so $x_1+x_2=-\frac{b}{a}$ and $x_1\cdot x_2=\frac{c}{a}$.
$$\frac{a}{c}=\frac{1}{x_1}\cdot \frac{1}{x_2}$$
$$\frac{1}{x_1}+\frac{1}{x_2}=\frac{x_1+x_2}{x_1x_2}=-\frac{b}{c}$$
and this shows that $\frac{1}{x_1}$ and $\frac{1}{x_2}$ are the roots of 
$$X^2+\frac{b}{c}X+\frac{a}{c}=0$$
Equivalently
$$cX^2+bX+a=0$$
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