3y(3y+6)=5y+11

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Solution for 3y(3y+6)=5y+11 equation:


Simplifying
3y(3y + 6) = 5y + 11

Reorder the terms:
3y(6 + 3y) = 5y + 11
(6 * 3y + 3y * 3y) = 5y + 11
(18y + 9y2) = 5y + 11

Reorder the terms:
18y + 9y2 = 11 + 5y

Solving
18y + 9y2 = 11 + 5y

Solving for variable 'y'.

Reorder the terms:
-11 + 18y + -5y + 9y2 = 11 + 5y + -11 + -5y

Combine like terms: 18y + -5y = 13y
-11 + 13y + 9y2 = 11 + 5y + -11 + -5y

Reorder the terms:
-11 + 13y + 9y2 = 11 + -11 + 5y + -5y

Combine like terms: 11 + -11 = 0
-11 + 13y + 9y2 = 0 + 5y + -5y
-11 + 13y + 9y2 = 5y + -5y

Combine like terms: 5y + -5y = 0
-11 + 13y + 9y2 = 0

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

Divide each side by '9'.
-1.222222222 + 1.444444444y + y2 = 0

Move the constant term to the right:

Add '1.222222222' to each side of the equation.
-1.222222222 + 1.444444444y + 1.222222222 + y2 = 0 + 1.222222222

Reorder the terms:
-1.222222222 + 1.222222222 + 1.444444444y + y2 = 0 + 1.222222222

Combine like terms: -1.222222222 + 1.222222222 = 0.000000000
0.000000000 + 1.444444444y + y2 = 0 + 1.222222222
1.444444444y + y2 = 0 + 1.222222222

Combine like terms: 0 + 1.222222222 = 1.222222222
1.444444444y + y2 = 1.222222222

The y term is 1.444444444y.  Take half its coefficient (0.722222222).
Square it (0.5216049380) and add it to both sides.

Add '0.5216049380' to each side of the equation.
1.444444444y + 0.5216049380 + y2 = 1.222222222 + 0.5216049380

Reorder the terms:
0.5216049380 + 1.444444444y + y2 = 1.222222222 + 0.5216049380

Combine like terms: 1.222222222 + 0.5216049380 = 1.74382716
0.5216049380 + 1.444444444y + y2 = 1.74382716

Factor a perfect square on the left side:
(y + 0.722222222)(y + 0.722222222) = 1.74382716

Calculate the square root of the right side: 1.32054048

Break this problem into two subproblems by setting 
(y + 0.722222222) equal to 1.32054048 and -1.32054048.

Subproblem 1

y + 0.722222222 = 1.32054048 Simplifying y + 0.722222222 = 1.32054048 Reorder the terms: 0.722222222 + y = 1.32054048 Solving 0.722222222 + y = 1.32054048 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.722222222' to each side of the equation. 0.722222222 + -0.722222222 + y = 1.32054048 + -0.722222222 Combine like terms: 0.722222222 + -0.722222222 = 0.000000000 0.000000000 + y = 1.32054048 + -0.722222222 y = 1.32054048 + -0.722222222 Combine like terms: 1.32054048 + -0.722222222 = 0.598318258 y = 0.598318258 Simplifying y = 0.598318258

Subproblem 2

y + 0.722222222 = -1.32054048 Simplifying y + 0.722222222 = -1.32054048 Reorder the terms: 0.722222222 + y = -1.32054048 Solving 0.722222222 + y = -1.32054048 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.722222222' to each side of the equation. 0.722222222 + -0.722222222 + y = -1.32054048 + -0.722222222 Combine like terms: 0.722222222 + -0.722222222 = 0.000000000 0.000000000 + y = -1.32054048 + -0.722222222 y = -1.32054048 + -0.722222222 Combine like terms: -1.32054048 + -0.722222222 = -2.042762702 y = -2.042762702 Simplifying y = -2.042762702

Solution

The solution to the problem is based on the solutions from the subproblems. y = {0.598318258, -2.042762702}

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