<s>[INST] How do I proceed with these quadratic equations? The question is 
$$ax^2 + bx + c=0 $$ and $$cx^2+bx+a=0$$ have a common root, if $b≠ a+c$, then what is $$a^3+b^3+c^3$$ 
 [/INST] If a=c, the equations become identical, we can hardly determine any relationship among a,b,c(=a).
If a≠c, 
if y is a root of $ax^2+bx+c=0$, then observe that $\frac{1}{y}$ is a root of $cx^2+bx+a=0$.
For the common root, $y=\frac{1}{y}=>y=±1$, but y=-1 makes a-b+c=0 ⇔ b=a+c which is not acceptable according to the given condition (as André Nicolas has observed).
So, y=1 if a≠c
=>a+b+c=0
=>a+b=-c
Cubing both sides, $(a+b)^3=(-c)^3$
$=>a^3+b^3+3ab(a+b)=-c^3$
or, $=>a^3+b^3+3ab(-c)=-c^3$
$=>a^3+b+c^3=3abc$ if a≠c.
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