[Math] How to find the equation with roots $\alpha/\beta$ and $\beta/\alpha$, given that $\alpha \ne \beta $, $ \alpha^2 = 5\alpha -3$, $\beta^2 = 5\beta -3$

Question

I have $\alpha \ne \beta $, $ \alpha^2 = 5\alpha -3$, $\beta^2 = 5\beta -3$.

I need to find out an equation whose roots are $\frac{\alpha}{\beta}$ and $\frac{\beta}{\alpha}$.

How to find out?

Answer 1

$\alpha+\beta=5$ and $\alpha\beta=3$.

Thus, $$\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=\frac{(\alpha+\beta)^2-2\alpha\beta}{\alpha\beta}=\frac{25-6}{3}=\frac{19}{3}$$ and $$\frac{\alpha}{\beta}\cdot\frac{\beta}{\alpha}=1,$$ which gives the answer: $$3x^2-19x+3=0.$$