Using the following equation, find the center and radius of the circle. You must show all work and calculations to receive credit.

x^2 − 4x + y^2 + 8y = −4

Question

Using the following equation, find the center and radius of the circle. You must show all work and calculations to receive credit.


x^2 − 4x + y^2 + 8y = −4

lilcathyx · Answer

i have no clue on how to do this. & i don't know the complete square thingy.

lilcathyx · Answer

@jim_thompson5910

jim_thompson5910 · Answer

focus on just x^2 - 4x

lilcathyx · Answer

ok

jim_thompson5910 · Answer

is x^2 - 4x a perfect square?

lilcathyx · Answer

yes ?

jim_thompson5910 · Answer

so you can factor x^2 - 4x into something of the form (a+b)^2 ?

lilcathyx · Answer

yes

jim_thompson5910 · Answer

what does it factor to?

lilcathyx · Answer

(x - 2) ^ 2 ?

jim_thompson5910 · Answer

let's check that

jim_thompson5910 · Answer

(x - 2) ^ 2 = (x-2)(x-2)

(x - 2) ^ 2 = x^2 - 2x - 2x + 4

(x - 2) ^ 2 = x^2 - 4x + 4

jim_thompson5910 · Answer

so x^2 - 4x isn't a perfect square because it's missing that +4 at the end

IF it had the +4 at the end, then x^2 - 4x + 4 is definitely a perfect square since it factors to (x-2)^2

jim_thompson5910 · Answer

what we can do is add 4 to both sides to fix that issue

x^2 - 4x + y^2 + 8y = -4

x^2 - 4x + y^2 + 8y+4 = -4+4 ... Add 4 to both sides.

(x^2 - 4x + 4) + y^2 + 8y = 0

jim_thompson5910 · Answer

we can then factor the stuff in the parenthesis to get (x-2)^2

so we go from

(x^2 - 4x + 4) + y^2 + 8y = 0

to

(x-2)^2 + y^2 + 8y = 0

making sense?

lilcathyx · Answer

a lil

lilcathyx · Answer

so now we do that to y^2 +8 = 0? @jim_thompson5910

lilcathyx · Answer

why not (x-2)^2 + 4 and then so on.

jim_thompson5910 · Answer

we have 

(x-2)^2 + y^2 + 8y = 0

jim_thompson5910 · Answer

y^2 + 8y is NOT a perfect square

(y+4)^2 = y^2 + 8y + 16

so we have to add 16 to both sides to allow us to factor

(x-2)^2 + y^2 + 8y = 0

(x-2)^2 + y^2 + 8y `+16` = 0 `+16`

(x-2)^2 + (y^2 + 8y +16) = 16

(x-2)^2 + (y + 4)^2 = 16

jim_thompson5910 · Answer

so in short, we go from 

x^2 - 4x + y^2 + 8y = -4

to

(x-2)^2 + y^2 + 8y = 0

and then that turns into

(x-2)^2 + (y + 4)^2 = 16

this is just a shorter version of the steps to give a summary in a way. Of course, look over each step carefully to make sure you know what's going on.

lilcathyx · Answer

okk. im going to look at every step you did.

jim_thompson5910 · Answer

ok don't hesitate to let me know where you get stuck if you do at all