x2+y2-10x+6y+27=0

Subtract 27 from both sides of the equation.

x2+y2-10x+6y=-27

Use the form ax2+bx+c, to find the values of a, b, and c.

a=1,b=-10,c=0

Consider the vertex form of a parabola.

a(x+d)2+e

Substitute the values of a and b into the formula d=b2a.

d=-102(1)

Simplify the right side.

Cancel the common factor of 10 and 2.

Factor 2 out of 10.

d=-2⋅52⋅1

Cancel the common factors.

Factor 2 out of 2⋅1.

d=-2⋅52(1)

Cancel the common factor.

d=-2⋅52⋅1

Rewrite the expression.

d=-51

Divide 5 by 1.

d=-1⋅5

d=-1⋅5

d=-1⋅5

Multiply -1 by 5.

d=-5

d=-5

Find the value of e using the formula e=c-b24a.

Simplify each term.

Raise -10 to the power of 2.

e=0-1004⋅1

Multiply 4 by 1.

e=0-1004

Divide 100 by 4.

e=0-1⋅25

Multiply -1 by 25.

e=0-25

e=0-25

Subtract 25 from 0.

e=-25

e=-25

Substitute the values of a, d, and e into the vertex form a(x+d)2+e.

(x-5)2-25

(x-5)2-25

Substitute (x-5)2-25 for x2-10x in the equation x2+y2-10x+6y=-27.

(x-5)2-25+y2+6y=-27

Move -25 to the right side of the equation by adding 25 to both sides.

(x-5)2+y2+6y=-27+25

Use the form ax2+bx+c, to find the values of a, b, and c.

a=1,b=6,c=0

Consider the vertex form of a parabola.

a(x+d)2+e

Substitute the values of a and b into the formula d=b2a.

d=62(1)

Cancel the common factor of 6 and 2.

Factor 2 out of 6.

d=2⋅32⋅1

Cancel the common factors.

Factor 2 out of 2⋅1.

d=2⋅32(1)

Cancel the common factor.

d=2⋅32⋅1

Rewrite the expression.

d=31

Divide 3 by 1.

d=3

d=3

d=3

Find the value of e using the formula e=c-b24a.

Simplify each term.

Raise 6 to the power of 2.

e=0-364⋅1

Multiply 4 by 1.

e=0-364

Divide 36 by 4.

e=0-1⋅9

Multiply -1 by 9.

e=0-9

e=0-9

Subtract 9 from 0.

e=-9

e=-9

Substitute the values of a, d, and e into the vertex form a(x+d)2+e.

(y+3)2-9

(y+3)2-9

Substitute (y+3)2-9 for y2+6y in the equation x2+y2-10x+6y=-27.

(x-5)2+(y+3)2-9=-27+25

Move -9 to the right side of the equation by adding 9 to both sides.

(x-5)2+(y+3)2=-27+25+9

Add -27 and 25.

(x-5)2+(y+3)2=-2+9

Add -2 and 9.

(x-5)2+(y+3)2=7

(x-5)2+(y+3)2=7

This is the form of a circle. Use this form to determine the center and radius of the circle.

(x-h)2+(y-k)2=r2

Match the values in this circle to those of the standard form. The variable r represents the radius of the circle, h represents the x-offset from the origin, and k represents the y-offset from origin.

r=7

h=5

k=-3

The center of the circle is found at (h,k).

Center: (5,-3)

These values represent the important values for graphing and analyzing a circle.

Center: (5,-3)

Radius: 7

Graph x^2+y^2-10x+6y+27=0