# Graph 2x^2+y^2+4y=144 2×2+y2+4y=144
Find the standard form of the ellipse.
Complete the square for y2+4y.
Use the form ax2+bx+c, to find the values of a, b, and c.
a=1,b=4,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=42(1)
Cancel the common factor of 4 and 2.
Factor 2 out of 4.
d=2⋅22⋅1
Cancel the common factors.
Factor 2 out of 2⋅1.
d=2⋅22(1)
Cancel the common factor.
d=2⋅22⋅1
Rewrite the expression.
d=21
Divide 2 by 1.
d=2
d=2
d=2
Find the value of e using the formula e=c-b24a.
Simplify each term.
Cancel the common factor of (4)2 and 4.
Factor 4 out of (4)2.
e=0-4⋅44(1)
Cancel the common factors.
Cancel the common factor.
e=0-4⋅44⋅1
Rewrite the expression.
e=0-41
Divide 4 by 1.
e=0-1⋅4
e=0-1⋅4
e=0-1⋅4
Multiply -1 by 4.
e=0-4
e=0-4
Subtract 4 from 0.
e=-4
e=-4
Substitute the values of a, d, and e into the vertex form a(x+d)2+e.
(y+2)2-4
(y+2)2-4
Substitute (y+2)2-4 for y2+4y in the equation 2×2+y2+4y=144.
2×2+(y+2)2-4=144
Move -4 to the right side of the equation by adding 4 to both sides.
2×2+(y+2)2=144+4
2×2+(y+2)2=148
Divide each term by 148 to make the right side equal to one.
2×2148+(y+2)2148=148148
Simplify each term in the equation in order to set the right side equal to 1. The standard form of an ellipse or hyperbola requires the right side of the equation be 1.
x274+(y+2)2148=1
x274+(y+2)2148=1
This is the form of an ellipse. Use this form to determine the values used to find the center along with the major and minor axis of the ellipse.
(x-h)2b2+(y-k)2a2=1
Match the values in this ellipse to those of the standard form. The variable a represents the radius of the major axis of the ellipse, b represents the radius of the minor axis of the ellipse, h represents the x-offset from the origin, and k represents the y-offset from the origin.
a=237
b=74
k=-2
h=0
The center of an ellipse follows the form of (h,k). Substitute in the values of h and k.
(0,-2)
Find c, the distance from the center to a focus.
Find the distance from the center to a focus of the ellipse by using the following formula.
a2-b2
Substitute the values of a and b in the formula.
(237)2-(74)2
Simplify.
Simplify the expression.
Apply the product rule to 237.
22372-(74)2
Raise 2 to the power of 2.
4372-(74)2
4372-(74)2
Rewrite 372 as 37.
Use axn=axn to rewrite 37 as 3712.
4(3712)2-(74)2
Apply the power rule and multiply exponents, (am)n=amn.
4⋅3712⋅2-(74)2
Combine 12 and 2.
4⋅3722-(74)2
Cancel the common factor of 2.
Cancel the common factor.
4⋅3722-(74)2
Divide 1 by 1.
4⋅371-(74)2
4⋅371-(74)2
Evaluate the exponent.
4⋅37-(74)2
4⋅37-(74)2
Multiply 4 by 37.
148-(74)2
Rewrite 742 as 74.
Use axn=axn to rewrite 74 as 7412.
148-(7412)2
Apply the power rule and multiply exponents, (am)n=amn.
148-7412⋅2
Combine 12 and 2.
148-7422
Cancel the common factor of 2.
Cancel the common factor.
148-7422
Divide 1 by 1.
148-741
148-741
Evaluate the exponent.
148-1⋅74
148-1⋅74
Simplify the expression.
Multiply -1 by 74.
148-74
Subtract 74 from 148.
74
74
74
74
Find the vertices.
The first vertex of an ellipse can be found by adding a to k.
(h,k+a)
Substitute the known values of h, a, and k into the formula.
(0,-2+237)
The second vertex of an ellipse can be found by subtracting a from k.
(h,k-a)
Substitute the known values of h, a, and k into the formula.
(0,-2-(237))
Simplify.
(0,-2-237)
Ellipses have two vertices.
Vertex1: (0,-2+237)
Vertex2: (0,-2-237)
Vertex1: (0,-2+237)
Vertex2: (0,-2-237)
Find the foci.
The first focus of an ellipse can be found by adding c to k.
(h,k+c)
Substitute the known values of h, c, and k into the formula.
(0,-2+74)
The first focus of an ellipse can be found by subtracting c from k.
(h,k-c)
Substitute the known values of h, c, and k into the formula.
(0,-2-(74))
Simplify.
(0,-2-74)
Ellipses have two foci.
Focus1: (0,-2+74)
Focus2: (0,-2-74)
Focus1: (0,-2+74)
Focus2: (0,-2-74)
Find the eccentricity.
Find the eccentricity by using the following formula.
a2-b2a
Substitute the values of a and b into the formula.
(237)2-(74)2237
Simplify.
Simplify the numerator.
Apply the product rule to 237.
22372-(74)2237
Raise 2 to the power of 2.
4372-(74)2237
Rewrite 372 as 37.
Use axn=axn to rewrite 37 as 3712.
4(3712)2-(74)2237
Apply the power rule and multiply exponents, (am)n=amn.
4⋅3712⋅2-(74)2237
Combine 12 and 2.
4⋅3722-(74)2237
Cancel the common factor of 2.
Cancel the common factor.
4⋅3722-(74)2237
Divide 1 by 1.
4⋅371-(74)2237
4⋅371-(74)2237
Evaluate the exponent.
4⋅37-(74)2237
4⋅37-(74)2237
Multiply 4 by 37.
148-(74)2237
Rewrite 742 as 74.
Use axn=axn to rewrite 74 as 7412.
148-(7412)2237
Apply the power rule and multiply exponents, (am)n=amn.
148-7412⋅2237
Combine 12 and 2.
148-7422237
Cancel the common factor of 2.
Cancel the common factor.
148-7422237
Divide 1 by 1.
148-741237
148-741237
Evaluate the exponent.
148-1⋅74237
148-1⋅74237
Multiply -1 by 74.
148-74237
Subtract 74 from 148.
74237
74237
Combine 74 and 37 into a single radical.
74372
Cancel the common factor of 74 and 37.
Factor 37 out of 74.
37⋅2372
Cancel the common factors.
Factor 37 out of 37.
37⋅237(1)2
Cancel the common factor.
37⋅237⋅12
Rewrite the expression.
212
Divide 2 by 1.
22
22
22
22
22
These values represent the important values for graphing and analyzing an ellipse.
Center: (0,-2)
Vertex1: (0,-2+237)
Vertex2: (0,-2-237)
Focus1: (0,-2+74)
Focus2: (0,-2-74)
Eccentricity: 22
Graph 2x^2+y^2+4y=144     