Rewrite general form of ellipse to standard form (what happened in step 3?)

Question

From my math book (Rewriting general form to standard from)

General form:

1. $$8y+4y^{2}-18x+9x^{2}=23$$
2. $$-18x+9x^2 +8y+4y^2 =23$$

What happened from step 2 to 3?

3. $$9(x-1)^2 -9+4(y+1)^2 -4=23$$
4. $$ 9(x-1)^2 +4(y+1)^2 =36$$
5. $$\frac{(x-1)^2}{4}+\frac{(y+1)^2}{9}$$

Answer 1

The standard form of an ellipse (and hyperbola) has terms of the form $\tfrac{(x-x_0)^2}{a^2}$ and $\tfrac{(y-x_0)^2}{b^2}$, so you'll want to rewrite "in that direction"; this is sometimes called completing the square.

2. $$\color{blue}{-18x+9x^2} +8y+4y^2 =23$$

What happened from step 2 to 3?

3. $$\color{green}{9(x-1)^2 -9}+4(y+1)^2 -4=23$$

Note that by expanding: $$\begin{align} \color{green}{9\left(x-1\right)^2 -9} & =9\left(x^2-2x+1\right)-9 \\ & =9x^2-18x+9-9 \\ & =\color{blue}{9x^2-18x} \end{align}$$

But then backwards! Maybe you can now figure it out for the terms in $y$?