I have a system of three quadratic equations in the unknowns $p,q,r$. There are parameters $a,b,c,x,y,z$. Can the solutions for $p,q,r$ be expressed as formulas containing $a,b,c,x,y,z$? The equations are:
$$(p^2+q^2+r^2-a)x=p(px+qy+rz)$$ $$(p^2+q^2+r^2-b)y=q(px+qy+rz)$$ $$(p^2+q^2+r^2-c)z=r(px+qy+rz)$$
Write the equations as $$(A-a)x=p B\tag 1$$ $$(A-b)y=q B\tag 2$$ $$(A-c)z=r B\tag 3$$ $$A=p^2+q^2+r^2\tag 4$$ $$B=px+qy+rz\tag 5$$
Using $(1)$, $(2)$ and $(3)$, we have $$p=\frac{x (A-a)}{B} \qquad \qquad q=\frac{y (A-b)}{B}\qquad \qquad r=\frac{z (A-c)}{B}\tag {***}$$ Plug in $(4)$ and $(5)$ to get $$A=\frac{x^2 (a-A)^2+y^2 (A-b)^2+z^2 (A-c)^2}{B^2}\tag 6$$ $$B=\frac{x^2 (A-a)}{B}+\frac{y^2 (A-b)}{B}+\frac{z^2 (A-c)}{B}\tag 7$$ So, two equations for two unknown variables $(A,B)$.
But $(7)$ gives $$B^2=A \left(x^2+y^2+z^2\right)-(a x^2+b y^2+c z^2) \tag 8$$ Plug in $(6)$ $$A=\frac{x^2 (a-A)^2+y^2 (A-b)^2+z^2 (A-c)^2}{A \left(x^2+y^2+z^2\right)-(a x^2+b y^2+c z^2)}\tag 9$$ Cross multiply and solve to obtain $$A=\frac{a^2 x^2+b^2 y^2+c^2 z^2}{a x^2+b y^2+c z^2} \tag {10}$$
Now, compute $B$ from $(8)$ and go back to $(***)$ to get $(p,q,r)$.
