Question
Solve the equation
x=−∣3y−y2∣9−3y,0<y<3x=−∣3y−y2∣9−3y,y>3x=−∣3y−y2∣9−3y,y<0x=∣3y−y2∣9−3y,0<y<3x=∣3y−y2∣9−3y,y>3x=∣3y−y2∣9−3y,y<0
Evaluate
(x2y2−1)×3=x2y3
Multiply the terms
3(x2y2−1)=x2y3
Rewrite the expression
3(y2x2−1)=y3x2
Expand the expression
More Steps

Evaluate
3(y2x2−1)
Apply the distributive property
3y2x2−3×1
Any expression multiplied by 1 remains the same
3y2x2−3
3y2x2−3=y3x2
Move the expression to the left side
3y2x2−3−y3x2=0
Collect like terms by calculating the sum or difference of their coefficients
(3y2−y3)x2−3=0
Move the constant to the right side
(3y2−y3)x2=3
Divide both sides
3y2−y3(3y2−y3)x2=3y2−y33
Divide the numbers
x2=3y2−y33
Take the root of both sides of the equation and remember to use both positive and negative roots
x=±3y2−y33
Simplify the expression
More Steps

Evaluate
3y2−y33
To take a root of a fraction,take the root of the numerator and denominator separately
3y2−y33
Simplify the radical expression
More Steps

Evaluate
3y2−y3
Factor the expression
y2(3−y)
The root of a product is equal to the product of the roots of each factor
y2×3−y
Reduce the index of the radical and exponent with 2
∣y∣×3−y
∣y∣×3−y3
Multiply by the Conjugate
∣y∣×3−y×3−y3×3−y
Calculate
∣y∣∣3−y∣3×3−y
Calculate
More Steps

Evaluate
3×3−y
The product of roots with the same index is equal to the root of the product
3(3−y)
Calculate the product
9−3y
∣y∣∣3−y∣9−3y
x=±∣y∣∣3−y∣9−3y
Separate the equation into 2 possible cases
x=∣y∣∣3−y∣9−3yx=−∣y∣∣3−y∣9−3y
Multiply the terms
x=∣y(3−y)∣9−3yx=−∣y∣∣3−y∣9−3y
Multiply the terms
x=∣y(3−y)∣9−3yx=−∣y(3−y)∣9−3y
Calculate
{x=−∣y(3−y)∣9−3y0<y<3{x=−∣y(3−y)∣9−3yy>3{x=−∣y(3−y)∣9−3yy<0{x=∣y(3−y)∣9−3y0<y<3{x=∣y(3−y)∣9−3yy>3{x=∣y(3−y)∣9−3yy<0
Simplify
x=−∣y(3−y)∣9−3y,0<y<3x=−∣y(3−y)∣9−3y,y>3x=−∣y(3−y)∣9−3y,y<0x=∣3y−y2∣9−3y,y<0x=∣y(3−y)∣9−3y,0<y<3x=∣y(3−y)∣9−3y,y>3
Simplify
x=−∣y(3−y)∣9−3y,0<y<3x=−∣y(3−y)∣9−3y,y>3x=−∣y(3−y)∣9−3y,y<0x=∣3y−y2∣9−3y,0<y<3x=∣3y−y2∣9−3y,y<0x=∣y(3−y)∣9−3y,y>3
Simplify
x=−∣y(3−y)∣9−3y,0<y<3x=−∣y(3−y)∣9−3y,y>3x=−∣y(3−y)∣9−3y,y<0x=∣3y−y2∣9−3y,0<y<3x=∣3y−y2∣9−3y,y>3x=∣3y−y2∣9−3y,y<0
Simplify
x=−∣3y−y2∣9−3y,y<0x=−∣y(3−y)∣9−3y,0<y<3x=−∣y(3−y)∣9−3y,y>3x=∣3y−y2∣9−3y,0<y<3x=∣3y−y2∣9−3y,y>3x=∣3y−y2∣9−3y,y<0
Simplify
x=−∣3y−y2∣9−3y,0<y<3x=−∣3y−y2∣9−3y,y<0x=−∣y(3−y)∣9−3y,y>3x=∣3y−y2∣9−3y,0<y<3x=∣3y−y2∣9−3y,y>3x=∣3y−y2∣9−3y,y<0
Solution
x=−∣3y−y2∣9−3y,0<y<3x=−∣3y−y2∣9−3y,y>3x=−∣3y−y2∣9−3y,y<0x=∣3y−y2∣9−3y,0<y<3x=∣3y−y2∣9−3y,y>3x=∣3y−y2∣9−3y,y<0
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
(x2y2−1)×3=x2y3
Multiply the terms
3(x2y2−1)=x2y3
To test if the graph of 3(x2y2−1)=x2y3 is symmetry with respect to the origin,substitute -x for x and -y for y
3((−x)2(−y)2−1)=(−x)2(−y)3
Evaluate
3(x2y2−1)=(−x)2(−y)3
Evaluate
More Steps

Evaluate
(−x)2(−y)3
Rewrite the expression
x2(−y3)
Use the commutative property to reorder the terms
−x2y3
3(x2y2−1)=−x2y3
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=6x−3yx2y2−6y
Calculate
(x2y2−1)3=x2y3
Simplify the expression
3(x2y2−1)=x2y3
Take the derivative of both sides
dxd(3(x2y2−1))=dxd(x2y3)
Calculate the derivative
More Steps

Evaluate
dxd(3(x2y2−1))
Simplify
3×dxd(x2y2−1)
Calculate
3(2xy2+2x2ydxdy)
3(2xy2+2x2ydxdy)=dxd(x2y3)
Calculate the derivative
More Steps

Evaluate
dxd(x2y3)
Use differentiation rules
dxd(x2)×y3+x2×dxd(y3)
Use dxdxn=nxn−1 to find derivative
2xy3+x2×dxd(y3)
Evaluate the derivative
More Steps

Evaluate
dxd(y3)
Use differentiation rules
dyd(y3)×dxdy
Use dxdxn=nxn−1 to find derivative
3y2dxdy
2xy3+3x2y2dxdy
3(2xy2+2x2ydxdy)=2xy3+3x2y2dxdy
Expand the expression
More Steps

Evaluate
3(2xy2+2x2ydxdy)
Apply the distributive property
3×2xy2+3×2x2ydxdy
Multiply the terms
6xy2+3×2x2ydxdy
Multiply the numbers
6xy2+6x2ydxdy
6xy2+6x2ydxdy=2xy3+3x2y2dxdy
Move the expression to the left side
6xy2+6x2ydxdy−3x2y2dxdy=2xy3
Move the expression to the right side
6x2ydxdy−3x2y2dxdy=2xy3−6xy2
Collect like terms by calculating the sum or difference of their coefficients
(6x2y−3x2y2)dxdy=2xy3−6xy2
Divide both sides
6x2y−3x2y2(6x2y−3x2y2)dxdy=6x2y−3x2y22xy3−6xy2
Divide the numbers
dxdy=6x2y−3x2y22xy3−6xy2
Solution
More Steps

Evaluate
6x2y−3x2y22xy3−6xy2
Rewrite the expression
6x2y−3x2y2xy(2y2−6y)
Rewrite the expression
xy(6x−3yx)xy(2y2−6y)
Reduce the fraction
6x−3yx2y2−6y
dxdy=6x−3yx2y2−6y
Show Solution

Find the second derivative
Find the second derivative with respect to x
Find the second derivative with respect to y
dx2d2y=72x2−108x2y+54y2x2−9y3x270y3−168y2−10y4+144y
Calculate
(x2y2−1)3=x2y3
Simplify the expression
3(x2y2−1)=x2y3
Take the derivative of both sides
dxd(3(x2y2−1))=dxd(x2y3)
Calculate the derivative
More Steps

Evaluate
dxd(3(x2y2−1))
Simplify
3×dxd(x2y2−1)
Calculate
3(2xy2+2x2ydxdy)
3(2xy2+2x2ydxdy)=dxd(x2y3)
Calculate the derivative
More Steps

Evaluate
dxd(x2y3)
Use differentiation rules
dxd(x2)×y3+x2×dxd(y3)
Use dxdxn=nxn−1 to find derivative
2xy3+x2×dxd(y3)
Evaluate the derivative
More Steps

Evaluate
dxd(y3)
Use differentiation rules
dyd(y3)×dxdy
Use dxdxn=nxn−1 to find derivative
3y2dxdy
2xy3+3x2y2dxdy
3(2xy2+2x2ydxdy)=2xy3+3x2y2dxdy
Expand the expression
More Steps

Evaluate
3(2xy2+2x2ydxdy)
Apply the distributive property
3×2xy2+3×2x2ydxdy
Multiply the terms
6xy2+3×2x2ydxdy
Multiply the numbers
6xy2+6x2ydxdy
6xy2+6x2ydxdy=2xy3+3x2y2dxdy
Move the expression to the left side
6xy2+6x2ydxdy−3x2y2dxdy=2xy3
Move the expression to the right side
6x2ydxdy−3x2y2dxdy=2xy3−6xy2
Collect like terms by calculating the sum or difference of their coefficients
(6x2y−3x2y2)dxdy=2xy3−6xy2
Divide both sides
6x2y−3x2y2(6x2y−3x2y2)dxdy=6x2y−3x2y22xy3−6xy2
Divide the numbers
dxdy=6x2y−3x2y22xy3−6xy2
Divide the numbers
More Steps

Evaluate
6x2y−3x2y22xy3−6xy2
Rewrite the expression
6x2y−3x2y2xy(2y2−6y)
Rewrite the expression
xy(6x−3yx)xy(2y2−6y)
Reduce the fraction
6x−3yx2y2−6y
dxdy=6x−3yx2y2−6y
Take the derivative of both sides
dxd(dxdy)=dxd(6x−3yx2y2−6y)
Calculate the derivative
dx2d2y=dxd(6x−3yx2y2−6y)
Use differentiation rules
dx2d2y=(6x−3yx)2dxd(2y2−6y)×(6x−3yx)−(2y2−6y)×dxd(6x−3yx)
Calculate the derivative
More Steps

Evaluate
dxd(2y2−6y)
Use differentiation rules
dxd(2y2)+dxd(−6y)
Evaluate the derivative
4ydxdy+dxd(−6y)
Evaluate the derivative
4ydxdy−6dxdy
dx2d2y=(6x−3yx)2(4ydxdy−6dxdy)(6x−3yx)−(2y2−6y)×dxd(6x−3yx)
Calculate the derivative
More Steps

Evaluate
dxd(6x−3yx)
Use differentiation rules
dxd(6x)+dxd(−3yx)
Evaluate the derivative
6+dxd(−3yx)
Evaluate the derivative
6−3y−3xdxdy
dx2d2y=(6x−3yx)2(4ydxdy−6dxdy)(6x−3yx)−(2y2−6y)(6−3y−3xdxdy)
Calculate
More Steps

Evaluate
(4ydxdy−6dxdy)(6x−3yx)
Use the the distributive property to expand the expression
4ydxdy×(6x−3yx)−6dxdy×(6x−3yx)
Multiply the terms
24yxdxdy−12y2xdxdy−6dxdy×(6x−3yx)
Multiply the terms
24yxdxdy−12y2xdxdy−36xdxdy+18yxdxdy
Calculate
42yxdxdy−12y2xdxdy−36xdxdy
dx2d2y=(6x−3yx)242yxdxdy−12y2xdxdy−36xdxdy−(2y2−6y)(6−3y−3xdxdy)
Calculate
More Steps

Evaluate
(2y2−6y)(6−3y−3xdxdy)
Use the the distributive property to expand the expression
(2y2−6y)×6+(2y2−6y)(−3y−3xdxdy)
Multiply the terms
12y2−36y+(2y2−6y)(−3y−3xdxdy)
Multiply the terms
12y2−36y−6y3−6y2xdxdy+18y2+18yxdxdy
Calculate
30y2−36y−6y3−6y2xdxdy+18yxdxdy
dx2d2y=(6x−3yx)242yxdxdy−12y2xdxdy−36xdxdy−(30y2−36y−6y3−6y2xdxdy+18yxdxdy)
Calculate
More Steps

Calculate
42yxdxdy−12y2xdxdy−36xdxdy−(30y2−36y−6y3−6y2xdxdy+18yxdxdy)
If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses
42yxdxdy−12y2xdxdy−36xdxdy−30y2+36y+6y3+6y2xdxdy−18yxdxdy
Subtract the terms
24yxdxdy−12y2xdxdy−36xdxdy−30y2+36y+6y3+6y2xdxdy
Add the terms
24yxdxdy−6y2xdxdy−36xdxdy−30y2+36y+6y3
dx2d2y=(6x−3yx)224yxdxdy−6y2xdxdy−36xdxdy−30y2+36y+6y3
Calculate
dx2d2y=3(2x−yx)28yxdxdy−2y2xdxdy−12xdxdy−10y2+12y+2y3
Use equation dxdy=6x−3yx2y2−6y to substitute
dx2d2y=3(2x−yx)28yx×6x−3yx2y2−6y−2y2x×6x−3yx2y2−6y−12x×6x−3yx2y2−6y−10y2+12y+2y3
Solution
More Steps

Calculate
3(2x−yx)28yx×6x−3yx2y2−6y−2y2x×6x−3yx2y2−6y−12x×6x−3yx2y2−6y−10y2+12y+2y3
Multiply the terms
More Steps

Multiply the terms
8yx×6x−3yx2y2−6y
Rewrite the expression
8yx×x(6−3y)2y2−6y
Cancel out the common factor x
8y×6−3y2y2−6y
Multiply the terms
6−3y8y(2y2−6y)
3(2x−yx)26−3y8y(2y2−6y)−2y2x×6x−3yx2y2−6y−12x×6x−3yx2y2−6y−10y2+12y+2y3
Multiply the terms
3(2x−yx)26−3y8y(2y2−6y)−6−3y2y2(2y2−6y)−12x×6x−3yx2y2−6y−10y2+12y+2y3
Multiply the terms
3(2x−yx)26−3y8y(2y2−6y)−6−3y2y2(2y2−6y)−2−y4(2y2−6y)−10y2+12y+2y3
Calculate the sum or difference
More Steps

Evaluate
6−3y8y(2y2−6y)−6−3y2y2(2y2−6y)−2−y4(2y2−6y)−10y2+12y+2y3
Rewrite the fractions
6−3y8y(2y2−6y)−6−3y2y2(2y2−6y)+−2+y4(2y2−6y)−10y2+12y+2y3
Reduce fractions to a common denominator
6−3y8y(2y2−6y)−6−3y2y2(2y2−6y)+(−2+y)(−3)4(2y2−6y)(−3)−3(−1)(y−2)10y2×3(−1)(y−2)+3(−1)(y−2)12y×3(−1)(y−2)+3(−1)(y−2)2y3×3(−1)(y−2)
Use the commutative property to reorder the terms
6−3y8y(2y2−6y)−6−3y2y2(2y2−6y)+−3(−2+y)4(2y2−6y)(−3)−3(−1)(y−2)10y2×3(−1)(y−2)+3(−1)(y−2)12y×3(−1)(y−2)+3(−1)(y−2)2y3×3(−1)(y−2)
Rewrite the expression
6−3y8y(2y2−6y)−6−3y2y2(2y2−6y)+−3(−2+y)4(2y2−6y)(−3)−−3(y−2)10y2×3(−1)(y−2)+3(−1)(y−2)12y×3(−1)(y−2)+3(−1)(y−2)2y3×3(−1)(y−2)
Rewrite the expression
6−3y8y(2y2−6y)−6−3y2y2(2y2−6y)+−3(−2+y)4(2y2−6y)(−3)−−3(y−2)10y2×3(−1)(y−2)+−3(y−2)12y×3(−1)(y−2)+3(−1)(y−2)2y3×3(−1)(y−2)
Rewrite the expression
6−3y8y(2y2−6y)−6−3y2y2(2y2−6y)+−3(−2+y)4(2y2−6y)(−3)−−3(y−2)10y2×3(−1)(y−2)+−3(y−2)12y×3(−1)(y−2)+−3(y−2)2y3×3(−1)(y−2)
Rewrite the expression
6−3y8y(2y2−6y)−6−3y2y2(2y2−6y)+6−3y4(2y2−6y)(−3)−6−3y10y2×3(−1)(y−2)+6−3y12y×3(−1)(y−2)+6−3y2y3×3(−1)(y−2)
Write all numerators above the common denominator
6−3y8y(2y2−6y)−2y2(2y2−6y)+4(2y2−6y)(−3)−10y2×3(−1)(y−2)+12y×3(−1)(y−2)+2y3×3(−1)(y−2)
Multiply the terms
6−3y16y3−48y2−2y2(2y2−6y)+4(2y2−6y)(−3)−10y2×3(−1)(y−2)+12y×3(−1)(y−2)+2y3×3(−1)(y−2)
Multiply the terms
6−3y16y3−48y2−(4y4−12y3)+4(2y2−6y)(−3)−10y2×3(−1)(y−2)+12y×3(−1)(y−2)+2y3×3(−1)(y−2)
Multiply the terms
6−3y16y3−48y2−(4y4−12y3)−24y2+72y−10y2×3(−1)(y−2)+12y×3(−1)(y−2)+2y3×3(−1)(y−2)
Multiply the terms
6−3y16y3−48y2−(4y4−12y3)−24y2+72y−(−30y3+60y2)+12y×3(−1)(y−2)+2y3×3(−1)(y−2)
Multiply the terms
6−3y16y3−48y2−(4y4−12y3)−24y2+72y−(−30y3+60y2)−36y2+72y+2y3×3(−1)(y−2)
Multiply the terms
6−3y16y3−48y2−(4y4−12y3)−24y2+72y−(−30y3+60y2)−36y2+72y−6y4+12y3
Calculate the sum or difference
6−3y70y3−168y2−10y4+144y
3(2x−yx)26−3y70y3−168y2−10y4+144y
Multiply by the reciprocal
6−3y70y3−168y2−10y4+144y×3(2x−yx)21
Multiply the terms
(6−3y)×3(2x−yx)270y3−168y2−10y4+144y
Use the commutative property to reorder the terms
3(6−3y)(2x−yx)270y3−168y2−10y4+144y
Expand the expression
More Steps

Evaluate
3(6−3y)(2x−yx)2
Expand the expression
3(6−3y)(4x2−4x2y+y2x2)
Multiply the terms
(18−9y)(4x2−4x2y+y2x2)
Apply the distributive property
18×4x2−18×4x2y+18y2x2−9y×4x2−(−9y×4x2y)−9y×y2x2
Multiply the numbers
72x2−18×4x2y+18y2x2−9y×4x2−(−9y×4x2y)−9y×y2x2
Multiply the numbers
72x2−72x2y+18y2x2−9y×4x2−(−9y×4x2y)−9y×y2x2
Multiply the numbers
72x2−72x2y+18y2x2−36yx2−(−9y×4x2y)−9y×y2x2
Multiply the terms
72x2−72x2y+18y2x2−36yx2−(−36y2x2)−9y×y2x2
Multiply the terms
72x2−72x2y+18y2x2−36yx2−(−36y2x2)−9y3x2
If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses
72x2−72x2y+18y2x2−36yx2+36y2x2−9y3x2
Subtract the terms
72x2−108x2y+18y2x2+36y2x2−9y3x2
Add the terms
72x2−108x2y+54y2x2−9y3x2
72x2−108x2y+54y2x2−9y3x270y3−168y2−10y4+144y
dx2d2y=72x2−108x2y+54y2x2−9y3x270y3−168y2−10y4+144y
Show Solution
