Why does solving this equation a certain way yield only complex roots instead of real ones? For the system of equations
$$
4x^2 + y = 4,\quad
x^4 - y = 1
$$
if I attempt to solve by solving each equation for $y$ and setting them equal to each other, I obtain
$$
4 - 4x^2 = x^4 - 1
$$
$$
-4(x^2 - 1) = (x^2 - 1)(x^2 + 1)
$$
$$
-4 = x^2 + 1
$$
$$
x = ±\sqrt5 i
$$
However, it can be shown by graphing the equations, and by following the method of elimination, that the system has the real solutions $(1,0)$ and $(-1,0)$.

Why is it that one method yields only complex roots while another yields the real roots?


You divided by $x^2-1$ in the third step. That expression is not necessarily non-zero..
