Please help, this should be simple, but I have a terrible headache. I need the answer, please.?

x² + y² + 2ax + 2by + c = 0 rearranges to (x+a)² + (y+b)² = a² + b² - c, a circle centered at (-a,-b) with a radius of √(a²+b²-c).

The centre of the circle x²+y²−8x+6y+21=0 is therefore (-(-8/2),-6/2), which simplifies to (4,-3).

x² - 8x + y² + 6y = - 21
[ x² - 8x + 16 ] + [ y² + 6y + 9 ] = - 21 + 25
[ x - 4 ]² + [y + 3]² = 4
C (4 , - 3)

x² + y² - 8x + 6y + 21 = 0
You need to put the equation into center-radius form, (x-h)² + (y-k)² = r², where (h,k) is the center and r is the radius.

regroup terms
(x²-8x) + (y²+6y) = -21

complete the squares (you NEED to learn how to do this)
(x²-8x+4²) + (y²+6y+3²) = -21 + 4² + 3²
(x-4)² + (y+3)² = 4
center of circle: (4,-3)
radius: √4 = 2

To understand "completing the square" observe how these example of perfect squares look when they are expanded and try to notice the pattern.
(x + 1)(x + 1) = x^2 + 2x + 1 In this example (2/2)^2 = 1
(x + 2)(x + 2) = x^2 + 4x + 4 In this example (4/2)^2 = 4
(x - 3)(x - 3) = x^2 - 6x + 9 In this example (-6/2)^2 = 9
Look at the number in front of the x in each example. It is called the coefficient of x. If you half that number and then square it you get the last number.

Using that idea, this is how we complete the square starting with x^2 – 8x
Hopefully you see that (-4)^2 = 16 is the number we need. To keep the expression unchanged we add 16 to complete the square but then subtract it.
x^2 – 8x = (x – 4)^2 – 16

Starting with y^2 + 6y you should see that (3)^2 = 9 is the number we need.
We add 9 to complete the square but to keep in balance we subtract it
y^2 + 6y = (y + 3)^2 - 9

Those are the two calculations you are supposed to know about, so that
(x^2 – 8x) + (y^2 + 6y) + 21 = 0 = (x – 4)^2 – 16 + (y + 3)^2 – 9 + 21
Gathering the constants and rearranging slightly, this becomes
(x – 4)^2 + (y + 3)^2 = 4

That should tell you that this circle has centre (4, -3) and radius 2

@@@@@@@@@@@@@@@@@@

Shortcut tip: You do not need to write all that out. When you see
x^2+y^2 − 8x + 6y + 21 = 0 just think -4 and +3 and calculate constants as
21 – (16 + 9) and you can immediately write
(x – 4)^2 + (y + 3)^2 – 4 = 0