please solve quadratic simultaneous equations. $x,y,z$ are real variables. $l,m,n$ are real positive constants.
Solve for real $x,y,z$:
$$\begin{align}
x^2+xy+y^2 &= l\\
y^2+yz+z^2 &= m\\
z^2+zx+x^2 &= n
\end{align}
$$
This problem has a relevance in Electrical Engineering.

$$x^2+xy+y^2=l~~(1),~ y^2+yz+z^2=m ~~~(2),~ z^2+zx+x^2=n~~~(3)$$
Subtract (1) from (2) and (1) from (3) to get
$$(x-y)(x+y+z)=m-l~~~(4),~ (x-z)(x+y+z)=n-l~~~(5)$$
Let $(x+y+z)=w$, then (4) and (5) give
$$(x-y)=\frac{m-l}{w}~~~(6), ~(x-z)=\frac{n-l}{w}~~~(7)$$
From these two we get $$[3x-(x+y+z)]w=m+n-2l \implies x=\frac{w^2+m+n-2l}{3w}~~~(8).$$
Similarly, $$y=\frac{w^2+l+b-2m}{3w}, ~~z=\frac{w^2+l+m-2n}{3w}~~~(9)$$
Putting these in Eq. (4), we get
$$w^4-(l+m+n)w^2+(l^2+m^2+n^2)-lm-mn-nl=0$$
This gives 
$$w^2=\frac{(l+m+n)\pm\sqrt{(l+m+n)^2-4[l^2+m^2+n^2-lm-mn-nl]}}{2}$$
$$w^2=\frac{(l+m+n) \pm \sqrt{3[2(lm+mn+nl)-(l^2+m^2+n^2)}}{2}~~~~(10)$$
Lastly, Eqs. (8), (9) and (10) give the values of $x,y,z$ in terms of$l,m,n.$. The roots will be real if $$l^2+m^2+n^2 \le 2(lm+mn+nl)$$
