Quadratic equation - change to zero

Question

if you have the following quadratic equation... how do you change 21 to 0?:

$x^2 – xy + y^2 = 21$

as far as i know i always have to change it to 0 if it isn't 0.
if so; do I add -21 after $y^2$ or add it to $y^ 2$ > $-21y^2$

Answer 1

If you want the right side to be zero, subtract $21$ from both sides, getting $$x^2-xy+y^2-21=0$$ You haven't said what you are trying to do with this equation. This would allow you to consider it as a quadratic in either $x$ or $y$ and use the quadratic formula to get $$y=\frac 12\left(x\pm\sqrt{x^2-4(x^2-21)}\right)$$

Answer 2

If you let $x\to x-a$ and $y\to y-b$, then you can get

$$(x-a)^2-(x-a)(y-b)+(y-b)^2=0$$

However, this comes at the following price,

$$a^2 – ab + b^2 = 21$$

So what have you gained?

Besides, this is an ellipse, what would it even mean if the right hand side is zero?