Circle and ellipse intersection points

Question

We have system of two equations:

$$x^2+y+y^2=4$$

$$x^2+xy+y^2=3$$

I can not find points of intersection of these two curves (just points of intersection) using basic construction tools - ruler and compass. Please help.

Answer 1

The intersections are the roots of a quartic. To see this, solve the first equation for $x^2$ and substitute into the second, giving $$4-y-y^2+y\sqrt{4-y-y^2}+y^2=3,$$ or $$y^2(4-y-y^2) = y-1.$$ Collecting terms and simplifying gives a quartic one of whose roots is $y=-1$. The remaining cubic is irreducible with three real roots.