(3y)(2y+7)=(6+11)

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Solution for (3y)(2y+7)=(6+11) equation: 


Simplifying
(3y)(2y + 7) = (6 + 11)

Remove parenthesis around (3y)
3y(2y + 7) = (6 + 11)

Reorder the terms:
3y(7 + 2y) = (6 + 11)
(7 * 3y + 2y * 3y) = (6 + 11)
(21y + 6y2) = (6 + 11)

Combine like terms: 6 + 11 = 17
21y + 6y2 = (17)
21y + 6y2 = 17

Solving
21y + 6y2 = 17

Solving for variable 'y'.

Reorder the terms:
-17 + 21y + 6y2 = 17 + -17

Combine like terms: 17 + -17 = 0
-17 + 21y + 6y2 = 0

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

Divide each side by '6'.
-2.833333333 + 3.5y + y2 = 0

Move the constant term to the right:

Add '2.833333333' to each side of the equation.
-2.833333333 + 3.5y + 2.833333333 + y2 = 0 + 2.833333333

Reorder the terms:
-2.833333333 + 2.833333333 + 3.5y + y2 = 0 + 2.833333333

Combine like terms: -2.833333333 + 2.833333333 = 0.000000000
0.000000000 + 3.5y + y2 = 0 + 2.833333333
3.5y + y2 = 0 + 2.833333333

Combine like terms: 0 + 2.833333333 = 2.833333333
3.5y + y2 = 2.833333333

The y term is 3.5y.  Take half its coefficient (1.75).
Square it (3.0625) and add it to both sides.

Add '3.0625' to each side of the equation.
3.5y + 3.0625 + y2 = 2.833333333 + 3.0625

Reorder the terms:
3.0625 + 3.5y + y2 = 2.833333333 + 3.0625

Combine like terms: 2.833333333 + 3.0625 = 5.895833333
3.0625 + 3.5y + y2 = 5.895833333

Factor a perfect square on the left side:
(y + 1.75)(y + 1.75) = 5.895833333

Calculate the square root of the right side: 2.428133714

Break this problem into two subproblems by setting 
(y + 1.75) equal to 2.428133714 and -2.428133714.


Subproblem 1
y + 1.75 = 2.428133714

Simplifying
y + 1.75 = 2.428133714

Reorder the terms:
1.75 + y = 2.428133714

Solving
1.75 + y = 2.428133714

Solving for variable 'y'.

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

Add '-1.75' to each side of the equation.
1.75 + -1.75 + y = 2.428133714 + -1.75

Combine like terms: 1.75 + -1.75 = 0.00
0.00 + y = 2.428133714 + -1.75
y = 2.428133714 + -1.75

Combine like terms: 2.428133714 + -1.75 = 0.678133714
y = 0.678133714

Simplifying
y = 0.678133714


Subproblem 2
y + 1.75 = -2.428133714

Simplifying
y + 1.75 = -2.428133714

Reorder the terms:
1.75 + y = -2.428133714

Solving
1.75 + y = -2.428133714

Solving for variable 'y'.

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

Add '-1.75' to each side of the equation.
1.75 + -1.75 + y = -2.428133714 + -1.75

Combine like terms: 1.75 + -1.75 = 0.00
0.00 + y = -2.428133714 + -1.75
y = -2.428133714 + -1.75

Combine like terms: -2.428133714 + -1.75 = -4.178133714
y = -4.178133714

Simplifying
y = -4.178133714


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {0.678133714, -4.178133714}

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