Help with Conics: Change General Form to Standard Form

[ms. confused]

Ok I seem to be having problems with changing the general form of a conic to standard form. I'm mainly confused with how to factor, since I haven't done it in a while, as well as how to go about completing the square.

Here's one of my problems:

2x^2 + y^2 + 12x – 2y + 15=0

I rearranged it to look like: 2x^2 + 12x + y^2 – 2y + 15=0

Then I "supposedly" completed the square:

(2x^2 + 12x +36) + (y^2 – 2y +1)= 22

Factoring is where I got stuck: 2(x^2 + 6x +18) + (y-1)^2= 22

I don't know what to do with what I got and the answer is supposed to be:

(x+3)^2 / 2 + (y-1)^2 / 4 = 1

 

[Hurkyl]

(2x^2 + 12x +36)

Your problem is that this isn't a square! (Though, x^2 + 12x + 36 is) Your problem is you need to factor out the two first, so that the coefficient on x^2 is a 1.

 

[ms. confused]

I did that and I got 2(x^2 + 6x +18).

 

[vitaly]

1. 2x^2 + 12x + 18 which is equivalent to 2(x+3)(x+3) + y^2 - 2y + 1 which is equivalnt to (y-1)(y-1) = -15 + 18 +1
2. Your equation is 2(x+3)^2 + (y-1)^2 = 4
3. Divide each side by 4. Now you have:
2(x+3)^2/4 + (y-1)^2/4 = 4/4
4. Now, your final product is:
(x+3)^2/2 + (y-1)^2/4 = 1

Is that the needed answer?

 

[Hurkyl]

You need to factor before you figure out the constant term. You picked 36, then factored, which is the wrong way around.

 

[ms. confused]

How did you get 2x^2 + 12x + 18? I got 2x^2 + 12x + 36.

 

[vitaly]

All you know is 2x^2 + 12x + ?.

Factor out the two to make it easy.

Now you have 2(x^2 +6x + ?)

Then you can fill in the square by making it 2(x^2 + 6x + 9) or 2(x+3)^2

 

[vitaly]

This should give you 2x^2 + 12x + 18.

I hope I'm doing this right...

 

[dextercioby]

If you know the answer,then u can cheat:
1.Make in the initial quadratic form the 2 substitutions
$$x\rightarrow u-3$$
$$v\rightarrow v+1$$

2.Show that the new quadratic form is
$$\frac{u^{2}}{2}+\frac{v^{2}}{4}=1$$

3.Reverse the substitution and find the answer.

Daniel.

 

[ms. confused]

Oh I see! Thanks for the help guys!