Question
Identify the conic
Find the standard equation of the circle
Find the radius of the circle
Find the center of the circle
(x−23)2+(y+1)2=433
Evaluate
x2+y2−3x+2y=5
Use the commutative property to reorder the terms
x2−3x+y2+2y=5
To complete the square, the same value needs to be added to both sides
x2−3x+49+y2+2y=5+49
Use a2−2ab+b2=(a−b)2 to factor the expression
(x−23)2+y2+2y=5+49
Add the numbers
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Evaluate
5+49
Reduce fractions to a common denominator
45×4+49
Reorder the terms
420+49
Add or subtract the terms
420+9
Calculate
429
(x−23)2+y2+2y=429
To complete the square, the same value needs to be added to both sides
(x−23)2+y2+2y+1=429+1
Use a2+2ab+b2=(a+b)2 to factor the expression
(x−23)2+(y+1)2=429+1
Solution
More Steps
Evaluate
429+1
Reduce fractions to a common denominator
429+44
Add or subtract the terms
429+4
Calculate
433
(x−23)2+(y+1)2=433
Show Solution
Solve the equation
Solve for x
Solve for y
x=23+29−4y2−8yx=23−29−4y2−8y
Evaluate
x2+y2−3x+2y=5
Rewrite the expression
x2+y2+2y−3x=5
Move the expression to the left side
x2+y2+2y−3x−5=0
Simplify
x2+y2+2y−5−3x=0
Rewrite in standard form
x2−3x+y2+2y−5=0
Substitute a=1,b=−3 and c=y2+2y−5 into the quadratic formula x=2a−b±b2−4ac
x=23±(−3)2−4(y2+2y−5)
Simplify the expression
More Steps
Evaluate
(−3)2−4(y2+2y−5)
Multiply the terms
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Evaluate
4(y2+2y−5)
Apply the distributive property
4y2+4×2y−4×5
Multiply
4y2+8y−4×5
Multiply the numbers
4y2+8y−20
(−3)2−(4y2+8y−20)
Rewrite the expression
32−(4y2+8y−20)
If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses
32−4y2−8y+20
Evaluate the power
9−4y2−8y+20
Add the terms
29−4y2−8y
x=23±29−4y2−8y
Solution
x=23+29−4y2−8yx=23−29−4y2−8y
Show Solution
Rewrite the equation
r=23cos(θ)−2sin(θ)+5cos2(θ)−6sin(2θ)+24r=23cos(θ)−2sin(θ)−5cos2(θ)−6sin(2θ)+24
Evaluate
x2+y2−3x+2y=5
To convert the equation to polar coordinates,substitute x for rcos(θ) and y for rsin(θ)
(cos(θ)×r)2+(sin(θ)×r)2−3cos(θ)×r+2sin(θ)×r=5
Factor the expression
(cos2(θ)+sin2(θ))r2+(−3cos(θ)+2sin(θ))r=5
Simplify the expression
r2+(−3cos(θ)+2sin(θ))r=5
Subtract the terms
r2+(−3cos(θ)+2sin(θ))r−5=5−5
Evaluate
r2+(−3cos(θ)+2sin(θ))r−5=0
Solve using the quadratic formula
r=23cos(θ)−2sin(θ)±(−3cos(θ)+2sin(θ))2−4×1×(−5)
Simplify
r=23cos(θ)−2sin(θ)±5cos2(θ)−6sin(2θ)+24
Solution
r=23cos(θ)−2sin(θ)+5cos2(θ)−6sin(2θ)+24r=23cos(θ)−2sin(θ)−5cos2(θ)−6sin(2θ)+24
Show Solution
Testing for symmetry
Testing for symmetry about the origin
Testing for symmetry about the x-axis
Testing for symmetry about the y-axis
Not symmetry with respect to the origin
Evaluate
x2+y2−3x+2y=5
To test if the graph of x2+y2−3x+2y=5 is symmetry with respect to the origin,substitute -x for x and -y for y
(−x)2+(−y)2−3(−x)+2(−y)=5
Evaluate
More Steps
Evaluate
(−x)2+(−y)2−3(−x)+2(−y)
Evaluate the power
x2+(−y)2−3(−x)+2(−y)
Evaluate the power
x2+y2−3(−x)+2(−y)
Multiply the numbers
x2+y2+3x+2(−y)
Multiply the numbers
x2+y2+3x−2y
x2+y2+3x−2y=5
Solution
Not symmetry with respect to the origin
Show Solution
Find the first derivative
Find the derivative with respect to x
Find the derivative with respect to y
dxdy=2y+2−2x+3
Calculate
x2+y2−3x+2y=5
Take the derivative of both sides
dxd(x2+y2−3x+2y)=dxd(5)
Calculate the derivative
More Steps
Evaluate
dxd(x2+y2−3x+2y)
Use differentiation rules
dxd(x2)+dxd(y2)+dxd(−3x)+dxd(2y)
Use dxdxn=nxn−1 to find derivative
2x+dxd(y2)+dxd(−3x)+dxd(2y)
Evaluate the derivative
More Steps
Evaluate
dxd(y2)
Use differentiation rules
dyd(y2)×dxdy
Use dxdxn=nxn−1 to find derivative
2ydxdy
2x+2ydxdy+dxd(−3x)+dxd(2y)
Evaluate the derivative
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Evaluate
dxd(−3x)
Use differentiation rule dxd(cf(x))=c×dxd(f(x))
−3×dxd(x)
Use dxdxn=nxn−1 to find derivative
−3×1
Any expression multiplied by 1 remains the same
−3
2x+2ydxdy−3+dxd(2y)
Evaluate the derivative
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Evaluate
dxd(2y)
Use differentiation rules
dyd(2y)×dxdy
Evaluate the derivative
2dxdy
2x+2ydxdy−3+2dxdy
2x+2ydxdy−3+2dxdy=dxd(5)
Calculate the derivative
2x+2ydxdy−3+2dxdy=0
Simplify
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Evaluate
2x+2ydxdy−3+2dxdy
Collect like terms by calculating the sum or difference of their coefficients
2x+(2y+2)dxdy−3
Rearrange the terms
2x−3+(2y+2)dxdy
2x−3+(2y+2)dxdy=0
Move the constant to the right side
(2y+2)dxdy=0−(2x−3)
Subtract the terms
More Steps
Evaluate
0−(2x−3)
Removing 0 doesn't change the value,so remove it from the expression
−(2x−3)
If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses
−2x+3
(2y+2)dxdy=−2x+3
Multiply both sides of the equation by 2y+21
(2y+2)dxdy×2y+21=(−2x+3)×2y+21
Multiply the terms
More Steps
Evaluate
(−2x+3)×2y+21
Rewrite the expression
−(2x−3)×2y+21
Calculate the product
2y+2−2x+3
(2y+2)dxdy×2y+21=2y+2−2x+3
Solution
dxdy=2y+2−2x+3
Show Solution
Graph