Common root of cubic and quadratic equation

Question

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.

This has a more complete answer in this post. – Tito Piezas III Mar 05 '19 at 09:51 — Tito Piezas III, Mar 05 '19 at 09:51

score 3 · Accepted Answer · edited Mar 05 '19 at 09:49

3

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.

edited Mar 05 '19 at 09:49

Tito Piezas III

60,745

answered Mar 05 '19 at 06:18

Ross Millikan

383,099

how can subtraction give common roots? Any logic behind there. I came accross people subtract Biquadratic and cubic for common root l... but why? – mathophile Jul 13 '20 at 00:35

Common root of cubic and quadratic equation

1 Answers1