I am looking for an analytical solution of the following quadratic simultaneous equations. $$(1)\ x^2-a_0(b_0-x-z)(c_0-y-x)=0$$ $$(2)\ y^2-a_1(b_1-y-x)(c_1-z-y)=0$$ $$(3)\ z^2-a_2(b_2-z-y)(c_2-x-z)=0$$ where $x$, $y$, and $z$ are variables; $a_i$, $b_i$, and $c_i$ ($i=0,1,2$) are constants.
I tried with Maple, but i couldn't get the solution...
I think that you are facing a monster !
You could reduce the problem to two equations $x,y$ eliminating $z$ from equation $(1)$. Replacing in $(2)$ and $(3)$ is just awful. You should end with a polynomial of degree $6$ in $x$ or $y$. Then, this explains that.
For illustration purposes, I used $a_i=i+1$, $b_i=a_i+1$, $c_i=b_i+1$ $(i=0,1,2)$. There are six real solutions $\{x_k,y_k,z_k\}$ $(k=1,2,\cdots,6)$, which express as functions of the roots of polynomial $$t^6-324 t^4+3312 t^3-12276 t^2+19008 t-10368=0$$