The roots of the equation $2x^2-3x+6=0$ are α and β. Find a quadratic equation with integral coefficients whose roots are $\frac{α}{β}$ and $\frac{β}{α}$.
The answer is $4x^2+5x+4=0$
I don't know how to get to the answer. Could someone explain the steps?
Best Answer
Rename $a= \alpha $ and $b=\beta$
So this is $(x-{a\over b})(x-{b\over a})=0$ thus
$$x^2-{a^2+b^2\over ab}x+1=0$$
Since $ab = {6\over 2}=3$ and $a^2+b^2 = (a+b)^2-2ab = ({3\over 2})^2-6 = {-15\over 4} $ we get:
$$x^2+{5\over 4}x+1=0$$
and thus the conclusion.