Question

Complete the square to rewrite the following equation. Identify the center and radius of the circle. You must show all work and calculations to receive credit.

x2 + 2x + y2 + 4y = 20

Answer 1

When you're completing the square, you're trying to make it in the form of a perfect square which is

\( a^2 + 2ab + b^2 = (a+b)^2\)

or

\( a^2 -2ab + b^2 = (a-b)^2 \)

Answer 2

Ok

Answer 3

So for your question, we have

\( (x^2 + 2x +\color{red}{ \text{__}}) + (y^2 + 2y +\color{orange}{ \text{__}}) = 20 + \color{red}{\text{__}}+ \color{orange}{ \text{__}}\)

so we're trying to find a number to add to \(x^2 + 2x\) so that way we can simplify it into something like (a+b)^2

and the same thing goes for \(y^2 + 4y\)

and to make sure that we're not changing the equation, we have to add both of those values to the other side of the equal sign as well

Does that make sense so far?

Answer 4

Yes

Answer 5

Excellent, so first let's find what we have to add to x^2 + 4x

\( a^2 + 2ab + b^2 = (a+b)^2\)

\( x^2 + 2x + \color{red}{\text{__}} = (x + ?)^2\)

Do you see how 'a' and 'x' are the same thing?

So we're trying to find that third term, but to do that we need to know what 'b' is

The middle term in the formula is 2ab

And in your question, we have 2x in the middle

We said that our 'a' is 'x'. Can you figure out what 'b' would have to be?

Answer 6

y

Answer 7

'B' = 'Y
'

Answer 8

No, we're trying to get your question to look like the formula so that way we can simplify it

We want the terms to be equal

2ab = 2x

and since we said that 'a' and 'x' are the same
(We can easily tell this because both start with a^2 and x^2)

so we get

2xb = 2x

what is the value of 'b'

Answer 9

is it 0

Answer 10

No

What does 'b' have to be for 2xb to be equal to 2x

Anything multiplied with 0 is going to be 0

2x * 0 = 0

2x * b = 2x

what is b?

if you divide 2x on both sides, what do you get?

Answer 11

1

Answer 12

b=1

Answer 13

Exactly!! b = 1

and so that last term is b^2

if you look in the formula, we're adding by b^2 so that way we can simplify it to (a+b)^2

so what is 1^2 = ?

Answer 14

1

Answer 15

Exactly! So now we have

\( a^2 + 2ab + b^2 = (a+b)^2\)

\( x^2 + 2x + \color{red}{\text{1}} = (x + 1)^2\)


Do you see how we did that?

We found out that b was 1, so we had to add b^2

and then we could simplify it into \((a+b)^2\)

Answer 16

So now we have

\( (x^2 + 2x +\color{red}{ \text{1}}) + (y^2 + 2y +\color{orange}{ \text{__}}) = 20 + \color{red}{\text{1}}+ \color{orange}{ \text{__}}\)

\( (x+1)^2~~~~~~~~~ + (y^2 + 2y +\color{orange}{ \text{__}}) = 20 + \color{red}{\text{1}}+ \color{orange}{ \text{__}}\)

Do you follow?

Answer 17

yes

Answer 18

i Do

Answer 19

So we just have to figure out what we add to \(y^2 + 4y\) and then we'll be all done

I apologize for my typo in my previous comments where I wrote `2y`
Your question has `4y`

So it's the same concept


\( a^2 + 2ab + b^2 = (a+b)^2\)

\( y^2 + 4y + \color{orange}{\text{__}} = (y + ?)^2\)

The middle term in the formula is 2ab

And in your question, we have 4y in the middle

Since the 'a' is 'y' can you figure out what 'b' would have to be?


2ab = 4y

a is the same y

2yb = 4b

what does b have to be?

Answer 20

2

Answer 21

Perfect, so what is b^2

and what can we simplify it all down to?

Answer 22

4

Answer 23

is it 4

Answer 24

yes, b^2 is 4

so we're adding 4

and remember that it gets simplified down to (a+b)^2

so 'b' is the term when we factor it

Answer 25

is it x2

Answer 26

It's just 2

Answer 27

Because

\( a^2 + 2ab + \color{orange}{b}^2 = (a + \color{orange}{b})^2 \)

and we found that b = 2 so

\( y^2 + 4y + \color{orange}{2}^2 = (a + \color{orange}{2})^2\)

Does that make sense?

Answer 28

yes

Answer 29

But remember, we're adding 2^2 on both sides of the equation!!!

And the equation gets simplified to (y+2)^2


\( (x+1)^2~~~~~~~~~ + (y^2 + 2y +\color{orange}{ \text{4}}) = 20 + \color{red}{\text{1}}+ \color{orange}{ \text{4}}\)


Do you see what I mean?

Since we added 4 to that equation, we have to add 4 to the other side

and then to complete it, we just have to factor down the \(y^2 + 4y + 4\) into \( (y+2)^2\)

and just add the numbers on the other side of the equal sign

Can you do that?

Answer 30

y^2+ 4y +8

Answer 31

no??

you just need to replace \(y^2 + 4y + 4\)

with \( (y+2)^2\)

remember all that work we did earlier??

Answer 32

\(\color{#0cbb34}{\text{Originally Posted by}}\) @AZ
Because

\( a^2 + 2ab + \color{orange}{b}^2 = (a + \color{orange}{b})^2 \)

and we found that b = 2 so

\( y^2 + 4y + \color{orange}{2}^2 = (y+ \color{orange}{2})^2\)

Does that make sense?
\(\color{#0cbb34}{\text{End of Quote}}\)

Answer 33

and then you just have to add up

20 + 1 + 4 = ??

Answer 34

Oh ok so it is 5

Answer 35

are you sure?

Answer 36

Thanks a lot for the help I really appreciate it.

Answer 37

oh I mean 25

Answer 38

You're most welcome!!

We haven't identified the center and the radius just yet, however

Answer 39

sorry about the 5 thing

Answer 40

ok

Answer 41

So now we have the equation in the form of

\( (x+1)^2+ (y+2)^2 = 25\)



And this is the standard form of a circle

\( (x-h)^2 + (y-k)^2 = r^2\)

(h, k) is going to be the center of your circle

and r is going to be the radius

Answer 42

ok

Answer 43

So how does one solve it

Answer 44

The numbers are all right there in the same form

Answer 45

Here's some colors to help you see it

\( (x+1)^2+ (y+2)^2 = \color{red}{25}\)


And this is the standard form of a circle

\( (x-h)^2 + (y-k)^2 = \color{red}{r^2}\)

(h, k) is going to be the center of your circle

r is your radius


so what is the radius?

if r^2 is equal to 25
what is r equal to??

Answer 46

5

Answer 47

Good, so that's your radius

Now what about the center of your circle

\( (x+1)^2+ (y+2)^2 = 25\)

\( (x-h)^2 + (y-k)^2 = r^2\)

(h, k) is going to be the center of your circle

be careful about the signs

we have (x-h)
there's a minus but the x-value of the center of the circle is 'h'

so in (x+1)

you have to flip the sign, the x-value isn't going to be 1 but -1 since it's x PLUS 1 and not x - 1

does that make sense?

Answer 48

somewhat

Answer 49

so what is the center?

I gave you the 'h' which is the x-value of the center

what is the 'k'

do the same thing

that's your y-value of the center

Answer 50

An image that might help you understand what it means by that (h, k) is the center

Answer 51

It is 4

Answer 52

but probably not

Answer 53

How?

Look at the numbers

Answer 54

\( (x\color{red}{+1})^2+ (y\color{orange}{+2})^2 = 25\)

\( (x \color{red}{-h})^2 + (y\color{orange}{-k})^2 = r^2\)

Answer 55

so what is 'h'

and what is 'k'

Answer 56

what does 'h' have to be to make x-h into x+1

what does 'k' have to be to make y-k into y+2

Answer 57

2
3

Answer 58

uhhhhh

those aren't even the numbers that's listed??

All you have to do is flip the sign

Answer 59

what I first thought but it did not make sense mutch though so,
-1
-2

Answer 60

so that's the h and k

and you write the center as (h, k)

Answer 61

(-1,-2)

Answer 62

is it that

Answer 63

Exactly!

Answer 64

so is it 5 and (-1,-2)

Answer 65

yes