<s>[INST] Can $ x^2+\frac{x^2}{(x+1)^2}=3 $ be solved by 'completing square method'? On using the quartic formula, there are two zeroes of this equation
$$
x^2+\frac{x^2}{(x+1)^2}=3
$$
So, I was wondering if this equation could be solved like a quadratic equation.
Is it possible. If yes, how?
 [/INST] We may notice that the golden ratio $\varphi=\frac{1+\sqrt{5}}{2}$ is a solution of the given equation, since $\varphi^2=\varphi+1$ implies
$$ \varphi^2+\frac{\varphi^2}{(\varphi+1)^2} = \varphi^2+\varphi^{-2} = L_2 = 3. $$
In particular, the algebraic conjugate of $\varphi$, given by $\bar{\varphi}=\frac{1-\sqrt{5}}{2}$, is also a solution of the given equation, and the polynomial $x^2-x-1$ (i.e. the minimal polynomial of $\varphi$) is a divisor of
$$ x^2(x+1)^2+x^2-3(x+1)^2.$$
By performing a polynomial division, we get that the other solutions are given by $x^2+3x+3=0$, i.e. by $x=\frac{-3\pm i\sqrt{3}}{2}$.
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