3(y+6)y=2(y+9)

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Solution for 3(y+6)y=2(y+9) equation:


Simplifying
3(y + 6) * y = 2(y + 9)

Reorder the terms:
3(6 + y) * y = 2(y + 9)

Reorder the terms for easier multiplication:
3y(6 + y) = 2(y + 9)
(6 * 3y + y * 3y) = 2(y + 9)
(18y + 3y2) = 2(y + 9)

Reorder the terms:
18y + 3y2 = 2(9 + y)
18y + 3y2 = (9 * 2 + y * 2)
18y + 3y2 = (18 + 2y)

Solving
18y + 3y2 = 18 + 2y

Solving for variable 'y'.

Reorder the terms:
-18 + 18y + -2y + 3y2 = 18 + 2y + -18 + -2y

Combine like terms: 18y + -2y = 16y
-18 + 16y + 3y2 = 18 + 2y + -18 + -2y

Reorder the terms:
-18 + 16y + 3y2 = 18 + -18 + 2y + -2y

Combine like terms: 18 + -18 = 0
-18 + 16y + 3y2 = 0 + 2y + -2y
-18 + 16y + 3y2 = 2y + -2y

Combine like terms: 2y + -2y = 0
-18 + 16y + 3y2 = 0

Begin completing the square.  Divide all terms by
3 the coefficient of the squared term: 

Divide each side by '3'.
-6 + 5.333333333y + y2 = 0

Move the constant term to the right:

Add '6' to each side of the equation.
-6 + 5.333333333y + 6 + y2 = 0 + 6

Reorder the terms:
-6 + 6 + 5.333333333y + y2 = 0 + 6

Combine like terms: -6 + 6 = 0
0 + 5.333333333y + y2 = 0 + 6
5.333333333y + y2 = 0 + 6

Combine like terms: 0 + 6 = 6
5.333333333y + y2 = 6

The y term is 5.333333333y.  Take half its coefficient (2.666666667).
Square it (7.111111113) and add it to both sides.

Add '7.111111113' to each side of the equation.
5.333333333y + 7.111111113 + y2 = 6 + 7.111111113

Reorder the terms:
7.111111113 + 5.333333333y + y2 = 6 + 7.111111113

Combine like terms: 6 + 7.111111113 = 13.111111113
7.111111113 + 5.333333333y + y2 = 13.111111113

Factor a perfect square on the left side:
(y + 2.666666667)(y + 2.666666667) = 13.111111113

Calculate the square root of the right side: 3.620926831

Break this problem into two subproblems by setting 
(y + 2.666666667) equal to 3.620926831 and -3.620926831.

Subproblem 1

y + 2.666666667 = 3.620926831 Simplifying y + 2.666666667 = 3.620926831 Reorder the terms: 2.666666667 + y = 3.620926831 Solving 2.666666667 + y = 3.620926831 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-2.666666667' to each side of the equation. 2.666666667 + -2.666666667 + y = 3.620926831 + -2.666666667 Combine like terms: 2.666666667 + -2.666666667 = 0.000000000 0.000000000 + y = 3.620926831 + -2.666666667 y = 3.620926831 + -2.666666667 Combine like terms: 3.620926831 + -2.666666667 = 0.954260164 y = 0.954260164 Simplifying y = 0.954260164

Subproblem 2

y + 2.666666667 = -3.620926831 Simplifying y + 2.666666667 = -3.620926831 Reorder the terms: 2.666666667 + y = -3.620926831 Solving 2.666666667 + y = -3.620926831 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-2.666666667' to each side of the equation. 2.666666667 + -2.666666667 + y = -3.620926831 + -2.666666667 Combine like terms: 2.666666667 + -2.666666667 = 0.000000000 0.000000000 + y = -3.620926831 + -2.666666667 y = -3.620926831 + -2.666666667 Combine like terms: -3.620926831 + -2.666666667 = -6.287593498 y = -6.287593498 Simplifying y = -6.287593498

Solution

The solution to the problem is based on the solutions from the subproblems. y = {0.954260164, -6.287593498}

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