Solve for x (quadratic equation)

Question

How do you solve the following equation:

\begin{align*} \sqrt{2017 + \sqrt{2017 - x}} &= x \end{align*}

I tried squaring it twice, but then I am left with quadratic equation that I can not solve.

Answer 1

after squaring we will get $$\sqrt{2017-x}=x^2-2017$$ squaring one more times we obtain $$0=x^4-4034x^2+x-2017+2017^2$$ the solution is given by $$x=\frac{1}{2} \left(1+\sqrt{8065}\right)$$

Answer 2

Let $2017-x=y^2$, where $y\geq0$.

Hence, $x+y>0$ and $2017+y=x^2$, which gives $y+x=x^2-y^2$ or $x-y=1$ and the rest is smooth. We have $y=x-1$, $2017+x-1=x^2$ or $$x^2-x-2016=0$$ and since $x>0$, we get the answer $\{\frac{1+\sqrt{8065}}{2}\}$.