Converting an equation based on square roots So I was wondering how to convert an equation of the form $\sqrt{x_1}+\sqrt{x_2}+\sqrt{x_3}+...\sqrt{x_n}+k=0$ into a polynomial equation based on each $x_i$. 
For example if the equation was $$\sqrt{x_1}+\sqrt{x_2}+k=0$$, then subtracting $\sqrt{x_2}$ from each side and squaring yields:$$x_1+k^2+2k\sqrt{x_1}=x_2.$$ 
This can then be rearranged to: $$2k\sqrt{x_1}=-x_1-k^2+x_2.$$
Squaring both sides yields: $$4k^2x_1=x_1^2+k^4+x_2^2+2k^2x_1-2x_1x_2-2k^2x_2.$$ 
Rearranging/simplifying yields:
 $$x_1^2+x_2^2+k^4-2k^2x_1-2x_1x_2-2k^2x_2 = 0.$$
How can I find an equation of this form given that $n$ is greater than $4$? I am most interested in when $n = 6$.

Just for fun, the polynomial drawn from $\sqrt a+\sqrt b+\sqrt c+k=0$, using the method by Somos:
$$a^4+6a^2b^2+6a^2c^2+6a^2k^4-4ab^3+4ab^2k^2+4abc^2-40abck^2+4abk^4-4ac^3+4ac^2k^2+4acb^2+4ack^4-4ak^6+b^4+6b^2c^2+6b^2k^4-4ba^3+4ba^2k^2-4bc^3+4bc^2k^2+4bca^2+4bck^4-4bk^6+c^4+6c^2k^4-4ca^3+4ca^2k^2-4cb^3+4cb^2k^2-4ck^6+k^8-4k^2a^3-4k^2b^3-4k^2c^3=0.$$
Then next challenge is to determine the number of terms as a function of the number of variables.
