How do you turn a radical system into a polynomial system?

Question

Suppose I have a system of equations \begin{align} y_1 &= \sqrt{(x + a_1)^2 + 1}\,,\\ y_2 &= \sqrt{(x + a_2)^2 + 1}\,,\\ &\vdots \end{align} I have read that it is always possible to re-express $x$ in terms of a new variable $u$ such that $y_1,y_2,\dots$ are polynomials in this new variable. I know of one explicit example, related to the one I've just written down \begin{align} y_1&=\sqrt{x - a_1}\,,\\ y_2&=\sqrt{x - a_2}\,. \end{align} This system is made into a polynomial system with the choice \begin{align} x = \frac{a_1 + a_2}{2} + \frac{a_1 - a_2}{2}\frac{1 + u^4}{2u^2}\,. \end{align} How might I go about constructing the relationship between $x$ and $u$ for the case I first laid out above? It seems like my expression for $x$ should have $N$ free parameters, one for each equation, although it is not clear to me what equations these coefficients should obey. Moreover, it is unclear to me how one should choose the specific polynomial in $u$.