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I would like to find the solutions $y$ to this type of equations: $$\left(1+x -\sqrt{(1+x)^2-4y}\right)^2=z$$ with conditions on $x,y,z$ (like $0\lt y\lt x\leq \frac18$ and $0\lt z\lt x^2$).

Using solve with the option to_poly_solve:

sage: solve((1+x - sqrt((1+x)^2-4*y))^2 == z, y, to_poly_solve=True)                                                 
[y == 1/2*x^2 - 1/2*(x + 1)*sqrt(x^2 + 2*x - 4*y + 1) + x - 1/4*z + 1/2]

does not seem to work because $y$ appears on the right side of the solution. I expect to find a solution like $$y=\frac14\left((1+x)^2-\left(1+x-\sqrt{z}\right)^2\right).$$

I also tried the same after specifying the conditions with assume(), without success.