<s>[INST] How to solve $x^{-2} + x^{-3} = \dfrac{x^3 + x^2}{4x^6}$? For online school I was given the equation $$x^{-2} + x^{-3} = \frac{x^3 + x^2}{4x^6}.$$ The problem lies in the addition of the bases, as I do not know how to solve that. Could you help me?
 [/INST] We have that
$$x^{-2} + x^{-3} = \dfrac{x^3 + x^2}{4x^6}$$
$$x^4+x^3=\frac{1}{4}(x^3+x^2)$$
$$x^4+\frac34 x^3-\frac14 x^2=0$$
$$x^2(4x^2+3x-1)=0$$
However, since $x=0$ is not a solution (it is in the denominator of the original equation), we can divide out by $x$ to get
$$4x^2+3x-1=0$$
This is a simple quadratic equation, can you take it from here?
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