Remark
I'm going to complete the square very quickly for all of these. You can fill in the missing steps.
Top One
(x^2 - 2x + 1) +(y^2 + 2y + 1) = 1 + 1 + 1
(x - 1)^2 + (y + 1)^2 = 3
Radius = sqrt(3)
Second One
Divide by 4 to start with
x^2 + y^2 + 4x + 6y - 10 = 0
(x^2 + 4x + 4) + (y^2 + 6y + 9) = 10 + 4 + 9
(x + 2)^2 + (y + 3)^2 = 23
Radius = sqrt(23)
Three
(x^2 - 4x + 4) + (y^2 + 4y + 4) = 10 + 8
(x - 2)^2 + (y + 2)^2 = 18
Radius = sqrt(18) You would be wise not to reduce this even though the reducation is 3sqrt(2). The comparison will be easier.
Four
(x^2 - 8x + 16) + (y^2 - 6y + 9) = 20 + 16 + 9
(x - 4)^2 + (y - 4)^2 = 44
Radius = sqrt(44)
Again don't reduce this
Five
(x^2 - 12x + 36) + (y^2 - 2y + 1) = 9 + 36 + 1
(x -6)^2 + (y - 1)^2 = 46
radius = sqrt(46)
Six
Divide through by 5
x^2 + y^2 - 4x + 6y + 8 = 0
(x^2 - 4x + 4) + (y^2 + 6y + 9) = - 8 + 4 + 9 = 5
(x -2)^2 + (y + 3)^2 = 5
Radius = sqrt(5)
Seven
Divide through by 2
x^2 + y^2 - 14x - 32y - 4 = 0
(x^2 - 14x + 49) + (y^2 - 32y + 256) = 4 + 49 + 256
(x - 7)^2 + (y - 16)^2 = 309
The radius is sqrt(309)
Comment
The calculation order produces
One . . . . . sqrt(3)
Two . . . . . . sqrt(23)
Three . . . . .sqrt(18)
Four . . . . . .sqrt(44)
Five . . . . . .sqrt(46)
Six . . . . . . .sqrt(5)
Seven . . . . sqrt(309)
I leave the ordering part to you.
