If equations $ax^3+2bx^2+3cx+4d=0$ and $ax^2+bx+c=0$ have a non zero common root, prove that $(c^2-2bd)(b^2-ac) \geq 0$.
I know the condition of common root of two quadratic equations but I have no idea on how to proceed with this question.
Common root of cubic and quadratic equation
common-rootcubicsquadraticsroots
Best Answer
You can multiply the second by $x$ and subtract from the first. That leaves $(2b-a)x^2+(3c-b)x+(4d-c)=0$ and $ax^2+bx+c=0$ as two quadratic equations, which you know how to handle.