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What are the conditions for the existence of real solutions for the following equations:

$$\begin{align} x^2&=a\cdot y+b\\ y^2&=c\cdot x+d\end{align}$$

where $a,b,c,d $ are real numbers.

These represent two parabolas; how might we find out the conditions for the existence of $0,2,4$ real solutions of the equations?

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    Don't forget the possibility of tangency.2011-12-19
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    Eliminating $y$ yields the (already) depressed quartic equation $x^4-2bx^2-a^2 cx+b^2-da^2$. [This](http://www.jstor.org/stable/2972804) gives a nice overview of the possibilities for the roots of such a quartic.2011-12-19
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    That link is great, but the proofs are missing, I suppose?2011-12-19
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    That's probably why I left it as a comment...2011-12-19

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