x2+y2-6x+8y=0

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)

Simplify the right side.

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=-1⋅3

d=-1⋅3

d=-1⋅3

Multiply -1 by 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.

(x-3)2-9

(x-3)2-9

Substitute (x-3)2-9 for x2-6x in the equation x2+y2-6x+8y=0.

(x-3)2-9+y2+8y=0

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

(x-3)2+y2+8y=0+9

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+y2-6x+8y=0.

(x-3)2+(y+4)2-16=0+9

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

(x-3)2+(y+4)2=0+9+16

Add 0 and 9.

(x-3)2+(y+4)2=9+16

Add 9 and 16.

(x-3)2+(y+4)2=25

(x-3)2+(y+4)2=25

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=5

h=3

k=-4

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

Center: (3,-4)

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

Center: (3,-4)

Radius: 5

Graph x^2+y^2-6x+8y=0