3y(2y+1)=8

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Solution for 3y(2y+1)=8 equation: 


Simplifying
3y(2y + 1) = 8

Reorder the terms:
3y(1 + 2y) = 8
(1 * 3y + 2y * 3y) = 8
(3y + 6y2) = 8

Solving
3y + 6y2 = 8

Solving for variable 'y'.

Reorder the terms:
-8 + 3y + 6y2 = 8 + -8

Combine like terms: 8 + -8 = 0
-8 + 3y + 6y2 = 0

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

Divide each side by '6'.
-1.333333333 + 0.5y + y2 = 0

Move the constant term to the right:

Add '1.333333333' to each side of the equation.
-1.333333333 + 0.5y + 1.333333333 + y2 = 0 + 1.333333333

Reorder the terms:
-1.333333333 + 1.333333333 + 0.5y + y2 = 0 + 1.333333333

Combine like terms: -1.333333333 + 1.333333333 = 0.000000000
0.000000000 + 0.5y + y2 = 0 + 1.333333333
0.5y + y2 = 0 + 1.333333333

Combine like terms: 0 + 1.333333333 = 1.333333333
0.5y + y2 = 1.333333333

The y term is 0.5y.  Take half its coefficient (0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
0.5y + 0.0625 + y2 = 1.333333333 + 0.0625

Reorder the terms:
0.0625 + 0.5y + y2 = 1.333333333 + 0.0625

Combine like terms: 1.333333333 + 0.0625 = 1.395833333
0.0625 + 0.5y + y2 = 1.395833333

Factor a perfect square on the left side:
(y + 0.25)(y + 0.25) = 1.395833333

Calculate the square root of the right side: 1.181453906

Break this problem into two subproblems by setting 
(y + 0.25) equal to 1.181453906 and -1.181453906.


Subproblem 1
y + 0.25 = 1.181453906

Simplifying
y + 0.25 = 1.181453906

Reorder the terms:
0.25 + y = 1.181453906

Solving
0.25 + y = 1.181453906

Solving for variable 'y'.

Move all terms containing y to the left, all other terms to the right.

Add '-0.25' to each side of the equation.
0.25 + -0.25 + y = 1.181453906 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + y = 1.181453906 + -0.25
y = 1.181453906 + -0.25

Combine like terms: 1.181453906 + -0.25 = 0.931453906
y = 0.931453906

Simplifying
y = 0.931453906


Subproblem 2
y + 0.25 = -1.181453906

Simplifying
y + 0.25 = -1.181453906

Reorder the terms:
0.25 + y = -1.181453906

Solving
0.25 + y = -1.181453906

Solving for variable 'y'.

Move all terms containing y to the left, all other terms to the right.

Add '-0.25' to each side of the equation.
0.25 + -0.25 + y = -1.181453906 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + y = -1.181453906 + -0.25
y = -1.181453906 + -0.25

Combine like terms: -1.181453906 + -0.25 = -1.431453906
y = -1.431453906

Simplifying
y = -1.431453906


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {0.931453906, -1.431453906}

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