<s>[INST] Solution of system of equations involving $x_{1},x_{2},x_{3}$ 
Solve for $x_{1},x_{2},x_{3}$, given
$ax^2_{1}+bx_{1}+c=x_{2}$
$ax^2_{2}+bx_{2}+c=x_{3}$
$ax^2_{3}+bx_{3}+c=x_{1}$

Try: from $(1)$ and $(2)$
$a(x^2_{1}-x^2_{2})+b(x_{1}-x_{2})=(x_{2} - x_{3})$
And from $(2)$ and $(3)$
$a(x^2_{2}-x^2_{3})+b(x_{2}-x_{3})=(x_{3}-x_{1})$
Could some help me to find $x_{1},x_{2},x_{3}$, Thanks
 [/INST] option 1:
$x_1 = x_2 = x_3$
$ax^2 + bx + c = x\\
ax^2 + (b-1)x + c = 0\\
x= \frac {-(b-1) \pm \sqrt {(b-1)^2 - 4ac}}{2a}\\
$
option 2:
$x_1 \ne x_2 \ne x_3$
Find a polynomial that intersects:
$(x_1,x_2),(x_2,x_3),(x_3,x_1)\\
p(x) = x_2 + (x-x_1)\frac {x_3-x_2}{x_2-x_1} + (x-x_1)(x-x_2)\left(\frac {x_1 - x_2}{(x_3-x_2)(x_3-x_1)} - \frac {x_2-x_1}{(x_3-x_2)(x_2-x_1)}\right)$
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