Find all possible values of a in system of equations 2

Question

How would you find all possible values of variable a for which the system of equations has a solution?

$ x^4+y^2=(a+\frac{1}{a})^3 $

$ x^4−y^2=(a-\frac{1}{a})^3$

Also, how would you prove that for any solution $(x,y)$ , $x^2+|y| \ge 4$ is true?

Answer 1

$$x^4\pm y^2=a^3\pm3a+\frac3a\pm\frac1{a^3}$$

so that

$$x^4=a^3+\frac3a$$ and

$$y^2=3a+\frac1{a^3}.$$

Both only require that $a>0$, and imply $x^4,y^2\ge4$ or $x^2,|y|\ge2$.

Answer 2

Hint: Adding both equations we obtain $$2x^4=\frac{2(3+a^4)}{a}$$