## Meehan98 one year ago Identify the standard form of the equation by completing the square. 3x^2 − 2y^2 − 12x − 16y + 4 = 0

1. Meehan98

I know how to begin by factoring the 3 and -2 so you're left with: 3(x^2-4x+4) -2(y+8y+16)= -4

2. Meehan98

And the next step is to make the constants equal so you get: 3(x^2-4) - 2(y+8)= -24

3. Jhannybean

I would start like this: group the x and y terms separately: $$(3x^2-12x) +(-2y^2-16y)+4=0$$ Then i'd send 4 to the other side so i have all my variables on one side and my constants on the other: $$(3x^2-12x)+(-2y^2+16y)=-4$$ Factor out the LCM for both sets of ( ) on the LHS : $$3(x^2-4x)-2(y^2+8y)=-4$$ The stuff inside the () is in the form $$(ax^2+bx)$$ So in order to complete the quadratic, we need to find new c values for both sets of ( ). We're goign to find $$c_1$$ which correlates to the first set of ( ) and $$c_2$$ which correlates to the second ( ) : $$c_1 = \left(\dfrac{-4}{2}\right)^2 = (-2)^2 = 4$$ $$c_2 = \left( \dfrac{8}{2}\right)^2 = (4)^2 = 16$$ Now we plug these in to the ( ) respectively : $$3(x^2-4x\color{red}{+4})-2(y^2+8y\color{blue}{+16})=-4$$ But now here's the problem child, you have to remember whatever you do on the LHS you also have to do on the RHS. because we pulled out the LCMs $$3$$ and $$-2$$, we're going to multiply both $$c_1$$ and $$c_2$$ with these LCMs and add them on the RHS: $$3(x^2-4x\color{red}{+4})-2(y^2+8y\color{blue}{+16}=-4\color{red}{+12} \color{blue}{-32}$$

4. Jhannybean

Now we just simplify both the LHS and RHS : $$3(x-2)^2 -2(y+4)^2 = -24$$ this completes the standard form of the equation: $$(x-h)^2+(y-k)^2=r^2$$

5. Jhannybean

$$3(x^2-4x\color{red}{+4})-2(y^2+8y\color{blue}{+16})=-4\color{red}{+12} \color{blue}{-32}$$* fixed a typo.

6. Meehan98

Okay, I just realized that I typed in a different answer than what I originally had, and I had the same answer as you and that's why I was confused because these are my choices for the correct answer.

7. Jhannybean

Okay, let me see, one minute.

8. Meehan98

Ah, I got it! (y+4)^2/12 - (x-2)^2/8 =1 because you divide -24 to both sides!

9. Jhannybean

Yeah :D

10. Jhannybean

It's asking for the elliptical form

11. Meehan98

You're amazing! Thanks for your help!!!!

12. Jhannybean

No problem. Did you understand my explanation, or were there any questions about it?

13. Meehan98

Nope, I understood it perfectly!

14. Jhannybean

Nice :D