Find the center and radius of x25xy26y32 Solutiongiven equat

Find the center and radius of: x^2+5x+y^2-6y=32

given equation is x^2 + 5x+y^2 -6y =32

now 5x can be written as 2*5/2* x

and 6y can be wrriten as 2*3y

x^2 +2*5/2x +y^2 -2*3y =32

now adding and subtracting 25/4 , adding and subtracting 9

x^2 +2*5/2x + 25/4 +y^2 -2*3y +9 -25/4 -9 =32

now x^2 +2*5/2*x +25/4 can be wrriten as (x+5/2)^2

by formule (a+b)^2 = a^2 +2ab +b^2 , here a=x , b=5/2

and y^2 -2*3*y+9 can be wrriten as (y-3)^2

by formule (a-b)^2 =a^2 -2ab +b^2 , here a=y , b=3

so the equation becomes

(x+5/2)^2 + (y-3)^2 -25/4 -9 =32

(x+5/2)^2 + (y-3)^2 = 32+9 +25/4

(x+5/2)^2 + (y-3)^2 = 41 +25/4

(x+5/2)^2 + (y-3)^2 = (164+25)/4

(x+5/2)^2 + (y-3)^2 =189/4

(x+5/2)^2 +(y-3)^2 = [3/2(sqrt(17) ) ]^2

this equation is of the circle form

(x-a)^2 + (y-b)^2 =c^2

where (a,b) is center and c=radiius

so center (a,b) = (-5/2 , 3)

radius = 3/2* sqrt(21)