x2+y2-14x+8y+53=0

Subtract 53 from both sides of the equation.

x2+y2-14x+8y=-53

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

a=1,b=-14,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=-142(1)

Simplify the right side.

Cancel the common factor of 14 and 2.

Factor 2 out of 14.

d=-2⋅72⋅1

Cancel the common factors.

Factor 2 out of 2⋅1.

d=-2⋅72(1)

Cancel the common factor.

d=-2⋅72⋅1

Rewrite the expression.

d=-71

Divide 7 by 1.

d=-1⋅7

d=-1⋅7

d=-1⋅7

Multiply -1 by 7.

d=-7

d=-7

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

Simplify each term.

Raise -14 to the power of 2.

e=0-1964⋅1

Multiply 4 by 1.

e=0-1964

Divide 196 by 4.

e=0-1⋅49

Multiply -1 by 49.

e=0-49

e=0-49

Subtract 49 from 0.

e=-49

e=-49

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

(x-7)2-49

(x-7)2-49

Substitute (x-7)2-49 for x2-14x in the equation x2+y2-14x+8y=-53.

(x-7)2-49+y2+8y=-53

Move -49 to the right side of the equation by adding 49 to both sides.

(x-7)2+y2+8y=-53+49

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

a=1,b=8,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=82(1)

Cancel the common factor of 8 and 2.

Factor 2 out of 8.

d=2⋅42⋅1

Cancel the common factors.

Factor 2 out of 2⋅1.

d=2⋅42(1)

Cancel the common factor.

d=2⋅42⋅1

Rewrite the expression.

d=41

Divide 4 by 1.

d=4

d=4

d=4

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

Simplify each term.

Raise 8 to the power of 2.

e=0-644⋅1

Multiply 4 by 1.

e=0-644

Divide 64 by 4.

e=0-1⋅16

Multiply -1 by 16.

e=0-16

e=0-16

Subtract 16 from 0.

e=-16

e=-16

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

(y+4)2-16

(y+4)2-16

Substitute (y+4)2-16 for y2+8y in the equation x2+y2-14x+8y=-53.

(x-7)2+(y+4)2-16=-53+49

Move -16 to the right side of the equation by adding 16 to both sides.

(x-7)2+(y+4)2=-53+49+16

Add -53 and 49.

(x-7)2+(y+4)2=-4+16

Add -4 and 16.

(x-7)2+(y+4)2=12

(x-7)2+(y+4)2=12

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=23

h=7

k=-4

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

Center: (7,-4)

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

Center: (7,-4)

Radius: 23

Graph x^2+y^2-14x+8y+53=0