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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.

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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.

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  • $\begingroup$ how can subtraction give common roots? Any logic behind there. I came accross people subtract Biquadratic and cubic for common root l... but why? $\endgroup$
    – mathophile
    Commented Jul 13, 2020 at 0:35

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