How do I complete the square of x^2 + xy + y^2 = 3?
Ive differentiated to find three equations
2x - 2ux - uy = 0
2y - ux - 2uy = 0
-(x^2 + xy + y^2 - 3) = 0
and need to find all solutions of x,y, and u. Ive solved for u and found x, y, u, are (1,1, 2/3) (-1,-1,2/3) but cannot find the other two solutions. Any help with that would be much appreciated, thought it had to do with completing the square but I guess not.
Solving for $x$ in the first equation gives $x = \dfrac{uy}{2(1-u)}$
Solving for $x$ in the second equation gives: $x = \dfrac{2y(1-u)}{u}$
Setting them equal gives $u^2y = 4y(1-u)^2$. So, if $y=0$, then $x=0$, which is not a solution to the third equation. Hence $y \neq 0$, allowing you to divide both sides by $y$. Taking the square root gives:
$u = \pm 2(1-u)$
$u = 2-2u$ or $u = 2u-2$ gives $u = 2$ or $u = \dfrac{2}{3}$.
If $u = \dfrac{2}{3}$, you found the two solutions. So, try $u = 2$:
If $u=2$, then $x=-y$ in both the first and second equations. Plugging this into the third equation:
$(-y)^2 +(-y)y + y^2-3 = y^2-y^2+y^2-3 = y^2-3 = 0$, so $y = \pm \sqrt{3}$. Hence, the other two answers are:
$(x,y,u) = (\sqrt{3},-\sqrt{3},2)$ or $(x,y,u) = (-\sqrt{3},\sqrt{3},2)$