<s>[INST] How to solve system of equations involving square roots How to solve the following system of equations? I've tried some basic techniques like adding/substracting and squaring but with no effect.
$$
\left\{ 
\begin{array}{c}
\sqrt{1 + x_1} + \sqrt{1 + x_2} + \sqrt{1 + x_3} + \sqrt{1 + x_4} = 2\sqrt{5} \\ 
\sqrt{1 - x_1} + \sqrt{1 - x_2} + \sqrt{1 - x_3} + \sqrt{1 - x_4} = 2\sqrt{3}
\end{array}
\right. 
$$
 [/INST] Let $\mathbf{r}_k=
\begin{pmatrix}
  \sqrt{1+x_{k}} \\ \sqrt{1-x_{k}}
\end{pmatrix}$, then $\displaystyle \sum_{k=1}^{4} \mathbf{r}_k=
\begin{pmatrix}
  2\sqrt{5} \\ 2\sqrt{3}
\end{pmatrix}$
Now,
 \begin{align*}
  \left| \sum_{k=1}^{4} \mathbf{r}_k \right| &=2\sqrt{5+3} \\
  &= 4\sqrt{2} \\
  \sum_{k=1}^{4} |\mathbf{r}_k| &= \sum_{k=1}^{4} \sqrt{2} \\
  &= 4\sqrt{2}
\end{align*}

Considering the inequality
  $$\left| \sum_{k=1}^{4} \mathbf{r}_k \right| \le 
  \sum_{k=1}^{4} |\mathbf{r}_k|$$
in which equality holds if and only if all $\mathbf{r}_{k}$ are equal.

That is
$$4
\begin{pmatrix}
  \sqrt{1+x_k} \\ \sqrt{1-x_k}
\end{pmatrix}=
\begin{pmatrix}
  2\sqrt{5} \\ 2\sqrt{3}
\end{pmatrix}$$
Hence $$\fbox{$x_1=x_2=x_3=x_4=\frac{1}{4}$}$$
</s>