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

Math
x2+y2-14x+8y+53=0
Subtract 53 from both sides of the equation.
x2+y2-14x+8y=-53
Complete the square for x2-14x.
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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.
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Cancel the common factor of 14 and 2.
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Factor 2 out of 14.
d=-2⋅72⋅1
Cancel the common factors.
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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.
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Simplify each term.
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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
Complete the square for y2+8y.
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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.
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Factor 2 out of 8.
d=2⋅42⋅1
Cancel the common factors.
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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.
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Simplify each term.
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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
Simplify -53+49+16.
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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

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