Let $x,y\in \mathbb{R}$ such that $(x^2-y^2+1)^2+4x^2y^2-x^2-y^2=0$. Find minimize and maximize of $$P=x^2+y^2$$
I just have $$(x^2-y^2+1)^2+4x^2y^2-x^2-y^2=0$$
Or $$4y^2=\left(x^2+y^2\right)^2+\left(x^2+y^2\right)+1$$
Or $$4y^2-(P^2+P+1)=0 (\text{*})$$
The equation (*) has root only $0^2-4\cdot 4\cdot \left(P^2+P+1\right)\ge 0$ then we found max and min of $P$ but it's wrong. Otherhand the equation occurs when $x\ne y $ so i cant use any inequalities. Help me
For your work: I got the maximum as $$\frac{1}{4}(1+\sqrt{5})^2$$ and this will attained for $$x=0,y=\frac{1}{2}(1+\sqrt{5})$$ The minimum is given by $$\frac{1}{4}(-1+\sqrt{5})$$ and will get for $$x=0,y=\frac{1}{2}(-1-\sqrt{5})$$