Question

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Answer 1

1. Take the x^2+12x and complete the square.
2. Take the y^2+8y and complete the square
3. Factor.
4. You will now recognize the type of conic it is.

Answer 2

So lets see if i understand this right. To complete the square you find x and y right?

Answer 3

I'll help you with the x part and then you can practice on the y part.

Answer 4

Ok thanks! :)

Answer 5

Take 1/2 of 12 which is the coefficient of x. That is 6. Square the 6. That is 36. Add 36 to both sides of the equation. Now we have:
\[x^2+12x+36+y^2+8y=48+36\]

Answer 6

Now you complete the square on the y^2 +8y

Answer 7

1/2 of 8 is 4. Square 4 = 16. So if im correct, we'd have x^2 + 12x + 36 + y^2 + 8y + 16 = 84 + 16?

Answer 8

(I added the 48 + 36)

Answer 9

Very good. Now you just built 2 trinomial squares on the left side. Factor them and add the 84+16 on the right

Answer 10

Im sorry to do this to you but im really bad at factoring. Can you help?

Answer 11

Can you factor anything? How about this: x^2+7x+12

Answer 12

So i have to find 2 numbers that add up to 7? That would be 4 + 3.

Answer 13

You have to find two numbers ...When you multiply them you get 12 When you add them you get 7. So yes. The numbers are 3 and 4 so what are the factors?

Answer 14

Ohhhhh ok. I just found something like this in my notebook. So you take x + 4 = 0 and x + 3 = 0. So x = -4 and x = -3.

Answer 15

Ok so i do the same thing with x^2 + 12x + 36 and y^2 + 8y + 16, right?

Answer 16

The factors are (x+4)(x+3) which if you use FOIL you get x^2+7x+12

Answer 17

Oh. Ok.

Answer 18

So in your problem you have x^2+12x+36
6x6 is 36
6+6=12
So the factors are (x+6)(x+6) which is (x+6)^2

Answer 19

Ok. And then y^2 + 8y + 16 would be (y + 4)(y + 4) or (y + 4)^2.

Answer 20

yes.

Answer 21

Ok and then add 84 + 16 and that = 100

Answer 22

So now the equation is:
\[(x+6)^2+(y+4)^2=100\]

Answer 23

Now which conic section is that. Your choices are parabola, circle, ellipse, hyperbola.

Answer 24

Is it an ellipse?

Answer 25

No. The equation of an ellipse looks like this:
\[\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1\]

Answer 26

Ok. The only other one that i thought it might be is a circle?

Answer 27

Yes. Very Good. It is a circle. You can always tell a circle because it has x^2 and y^2 and the coefficients are the same.

Answer 28

Yay! Awesome :)

Answer 29

Now you can answer the other questions about it.

Answer 30

Ok im so sorry, im so brain dead. Can you help me find the domain and range and the center. I got the axes of symmetry already.

Answer 31

Oh. Is the center (-6, -4)?

Answer 32

@Mertsj

Answer 33

yes it is. Very good.

Answer 34

What did you get for the axes of symmetry?

Answer 35

x = -6 and y = -4 (axes of symmetry)

Answer 36

The center is (-6,-4) as you said and the radius is 10. So the domain is 10 units right and 10 units left of the center. So the domain is [-16,4]. The range is 10 units above and below the center. Can you do it?

Answer 37

Every diameter of a circle is an axis of symmetry so there is an infinite number of them.

Answer 38

Yeah i think so. Hold on...

Answer 39

Wouldnt it be the same thing, or is it (4, -16)?

Answer 40

No no no no hold on... i think i have it now...

Answer 41

That domain is in interval notation. It is the set of x values. It is not an ordered pair. Another way to write it would be
\[-16\le x \le4\]

Answer 42

Is it (6, -14)?

Answer 43

Yes, it would be from -14 to 6....you should put the smaller number first. Also the endpoints of the interval are included so you should use brackets, not parentheses...[-14,6]

Answer 44

Ok so the Domain is [-16, 4] and the Range is [-14, 6]. Thank you sooooo much! You were so patient!

Answer 45

Yes. Good job. You're welcome.