3(5.5+2y)+2y^2=77

Solution for 3(5.5+2y)+2y^2=77 equation:

Simplifying
3(5.5 + 2y) + 2y2 = 77
(5.5 * 3 + 2y * 3) + 2y2 = 77
(16.5 + 6y) + 2y2 = 77

Solving
16.5 + 6y + 2y2 = 77

Solving for variable 'y'.

Reorder the terms:
16.5 + -77 + 6y + 2y2 = 77 + -77

Combine like terms: 16.5 + -77 = -60.5
-60.5 + 6y + 2y2 = 77 + -77

Combine like terms: 77 + -77 = 0
-60.5 + 6y + 2y2 = 0

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

Divide each side by '2'.
-30.25 + 3y + y2 = 0

Move the constant term to the right:

Add '30.25' to each side of the equation.
-30.25 + 3y + 30.25 + y2 = 0 + 30.25

Reorder the terms:
-30.25 + 30.25 + 3y + y2 = 0 + 30.25

Combine like terms: -30.25 + 30.25 = 0.00
0.00 + 3y + y2 = 0 + 30.25
3y + y2 = 0 + 30.25

Combine like terms: 0 + 30.25 = 30.25
3y + y2 = 30.25

The y term is 3y.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3y + 2.25 + y2 = 30.25 + 2.25

Reorder the terms:
2.25 + 3y + y2 = 30.25 + 2.25

Combine like terms: 30.25 + 2.25 = 32.5
2.25 + 3y + y2 = 32.5

Factor a perfect square on the left side:
(y + 1.5)(y + 1.5) = 32.5

Calculate the square root of the right side: 5.700877125

Break this problem into two subproblems by setting 
(y + 1.5) equal to 5.700877125 and -5.700877125.

Subproblem 1
y + 1.5 = 5.700877125

Simplifying
y + 1.5 = 5.700877125

Reorder the terms:
1.5 + y = 5.700877125

Solving
1.5 + y = 5.700877125

Solving for variable 'y'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + y = 5.700877125 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + y = 5.700877125 + -1.5
y = 5.700877125 + -1.5

Combine like terms: 5.700877125 + -1.5 = 4.200877125
y = 4.200877125

Simplifying
y = 4.200877125

Subproblem 2
y + 1.5 = -5.700877125

Simplifying
y + 1.5 = -5.700877125

Reorder the terms:
1.5 + y = -5.700877125

Solving
1.5 + y = -5.700877125

Solving for variable 'y'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + y = -5.700877125 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + y = -5.700877125 + -1.5
y = -5.700877125 + -1.5

Combine like terms: -5.700877125 + -1.5 = -7.200877125
y = -7.200877125

Simplifying
y = -7.200877125

Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {4.200877125, -7.200877125}