Non-zero constant roots of an equation
After working out a couple of similar questions, I came across this particular which left me blank, I kindly ask for any hints on how to tackle this problem.
The roots of the equation $x^2 + px + q = 0$ are $\alpha$ and $\beta$, where $p$ and $q$ are non-zero constants. Find the expressions for $\frac{1}{\alpha}$+$\frac{1}{\beta}$ and $\frac{\alpha }{\beta}$+$\frac{\beta}{\alpha}$ in terms of $p$ and $q$. Hence, or otherwise, write down the equation whose roots are: $$\alpha + \frac{1}{\alpha} \beta + \frac{1}{\beta}$$.
The question itself is understandable, the part that boggles me right now is the root that he gave you. Should I proceed as I normally* would and replace Sum of Roots with $p$ and Product of Roots with $q$?
*I would normally get the Sum and Product of the roots that is given to me(most of the time was a complex number) and substitute it as: $$x^2 + (Sum of Roots)x + (Product of Roots) = 0$$
However this method seems completely off from what I have been asked to do. Thank you for your time.
$\endgroup$1 Answer
$\begingroup$Using Vieta's formula, $\alpha+\beta=-\dfrac p1,\alpha\beta=\dfrac q1$
$\dfrac1\alpha+\dfrac1\beta=\dfrac{\alpha+\beta}{\alpha\beta}$
$\dfrac\beta\alpha+\dfrac\alpha\beta=\dfrac{\alpha^2+\beta^2}{\alpha\beta}=\dfrac{(\alpha+\beta)^2-2\alpha\beta}{\alpha\beta}$
$\endgroup$
