Here is an interesting problem. Perhaps someone would be so kind as to give me a shove in the right direction?.
If $ax^{3}+3bx^{2}+3cx+d$ and $ax^{2}+2bx+c$ share a common root, then prove that $(ac-b^{2})(bd-c^{2})\geq 0$
I thought about equating coefficients somehow, but that got messy.
I used the quadratic formula on the given quadratic to find
$\displaystyle x=\frac{-b\pm\sqrt{b^{2}-ac}}{a}$ are two roots.
So, if the cubic shares on of these, then I should be able to sub this in for x in the cubic.
Upon doing so, I got:
$\frac{2(ac-b^{2})\sqrt{b^{2}-4ac}}{a^{2}}-\frac{3bc}{a}+\frac{2b^{3}}{a^{2}}+d$
This is where I got hung up. This may not even by a good way to go about it.
I see a part of what is to be proven in the above expression, $ac-b^{2}$
Setting it to 0 does not help much.
I also thought about dividing them. If they share a root, then it should reduce to a quadratic in the numerator and a linear denominator. But, then what?.
Can anyone give a hint as to the best way to proceed?.