3y^2+[6y^2+(3y-4)]=0

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Solution for 3y^2+[6y^2+(3y-4)]=0 equation:


Simplifying
3y2 + [6y2 + (3y + -4)] = 0

Reorder the terms:
3y2 + [6y2 + (-4 + 3y)] = 0

Remove parenthesis around (-4 + 3y)
3y2 + [6y2 + -4 + 3y] = 0

Reorder the terms:
3y2 + [-4 + 3y + 6y2] = 0

Remove brackets around [-4 + 3y + 6y2]
3y2 + -4 + 3y + 6y2 = 0

Reorder the terms:
-4 + 3y + 3y2 + 6y2 = 0

Combine like terms: 3y2 + 6y2 = 9y2
-4 + 3y + 9y2 = 0

Solving
-4 + 3y + 9y2 = 0

Solving for variable 'y'.

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

Divide each side by '9'.
-0.4444444444 + 0.3333333333y + y2 = 0

Move the constant term to the right:

Add '0.4444444444' to each side of the equation.
-0.4444444444 + 0.3333333333y + 0.4444444444 + y2 = 0 + 0.4444444444

Reorder the terms:
-0.4444444444 + 0.4444444444 + 0.3333333333y + y2 = 0 + 0.4444444444

Combine like terms: -0.4444444444 + 0.4444444444 = 0.0000000000
0.0000000000 + 0.3333333333y + y2 = 0 + 0.4444444444
0.3333333333y + y2 = 0 + 0.4444444444

Combine like terms: 0 + 0.4444444444 = 0.4444444444
0.3333333333y + y2 = 0.4444444444

The y term is 0.3333333333y.  Take half its coefficient (0.1666666667).
Square it (0.02777777779) and add it to both sides.

Add '0.02777777779' to each side of the equation.
0.3333333333y + 0.02777777779 + y2 = 0.4444444444 + 0.02777777779

Reorder the terms:
0.02777777779 + 0.3333333333y + y2 = 0.4444444444 + 0.02777777779

Combine like terms: 0.4444444444 + 0.02777777779 = 0.47222222219
0.02777777779 + 0.3333333333y + y2 = 0.47222222219

Factor a perfect square on the left side:
(y + 0.1666666667)(y + 0.1666666667) = 0.47222222219

Calculate the square root of the right side: 0.687184271

Break this problem into two subproblems by setting 
(y + 0.1666666667) equal to 0.687184271 and -0.687184271.

Subproblem 1

y + 0.1666666667 = 0.687184271 Simplifying y + 0.1666666667 = 0.687184271 Reorder the terms: 0.1666666667 + y = 0.687184271 Solving 0.1666666667 + y = 0.687184271 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.1666666667' to each side of the equation. 0.1666666667 + -0.1666666667 + y = 0.687184271 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + y = 0.687184271 + -0.1666666667 y = 0.687184271 + -0.1666666667 Combine like terms: 0.687184271 + -0.1666666667 = 0.5205176043 y = 0.5205176043 Simplifying y = 0.5205176043

Subproblem 2

y + 0.1666666667 = -0.687184271 Simplifying y + 0.1666666667 = -0.687184271 Reorder the terms: 0.1666666667 + y = -0.687184271 Solving 0.1666666667 + y = -0.687184271 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.1666666667' to each side of the equation. 0.1666666667 + -0.1666666667 + y = -0.687184271 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + y = -0.687184271 + -0.1666666667 y = -0.687184271 + -0.1666666667 Combine like terms: -0.687184271 + -0.1666666667 = -0.8538509377 y = -0.8538509377 Simplifying y = -0.8538509377

Solution

The solution to the problem is based on the solutions from the subproblems. y = {0.5205176043, -0.8538509377}

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