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I have a rather messy pair of simultaneous equations, which I need to solve for x: $\left(x+2n-1\over2\right)^2+\left(\sqrt{1-\left(x^2-2\over2\right)^2}+\sqrt{1-\left(-x^2+x+2n+1\over2\right)^2}\right)^2=x^2$$\left(x^2-x-1\over2\right)^2+\left(\sqrt{1-n^2}+\sqrt{1-\left(-x^2+x+2n+1\over2\right)^2}\right)^2=x^2$

The only information I know about $n$ and $x$ is that there is a solution where both are real numbers, $0<n<1$ and $\sqrt{2+\sqrt2}<x<2$, assuming my calculations so far have been correct. I only need to know the value of $x$.

The method that I would use to solve this is to rearrange both equations to make $n$ the subject, and then since they are equal to each other I would solve for $x$. The problem however is that I anticipate that the resulting equation would be a relatively high degree polynomial function of $x$, which I may have trouble finding the roots of.

If anyone could tell me a better way to solve this, or show me their workings of how to solve this, I would be very grateful.

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I would start by substituting one $x^2$ into the other equation and multiplying out the squared brackets. Then subtract one equation from the other. You will end up with two new equations which may well be simpler to combine and solve for $x$ or $n$.

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