Invalid subtraction when solving system of equations? I'm trying to solve these two equations:
$$\begin{cases}
1-4x(x^2+y^2)=0 \\
1-4y(x^2+y^2)=0
\end{cases}$$
and I tried to do it by subtracting the first equation from the second, yielding $(4x-4y)(x^2+y^2)=0$. Clearly this is satisfied when $x=y$, which gives $(x,y)=(\frac{1}{2},\frac{1}{2})$, or when $x^2+y^2=0$, which gives $(x,y)=(0,0)$.
But when I plug them back in, it's obvious that $(x,y)=(0,0)$ is not a solution. Where did I go wrong?

First $x=y$ yields $8x^3=1$
$x=\dfrac{1}{2},\dfrac{-1+\sqrt3i}{4},\dfrac{-1-\sqrt3i}{4}$
$\therefore(x,y)=\left(\dfrac{1}{2},\dfrac{1}{2}\right),\left(\dfrac{-1+\sqrt3i}{4},\dfrac{-1+\sqrt3i}{4}\right),\left(\dfrac{-1-\sqrt3i}{4},\dfrac{-1-\sqrt3i}{4}\right)$
But $x^2+y^2=0$ yields $1=0$ which is not useful.
Note than $x^2+y^2=0$ should not immediately implies $x=y=0$ , which should have no concept abut the false solution $(x,y)=(0,0)$ .
