3y(2y+7)=8

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Solution for 3y(2y+7)=8 equation: 


Simplifying
3y(2y + 7) = 8

Reorder the terms:
3y(7 + 2y) = 8
(7 * 3y + 2y * 3y) = 8
(21y + 6y2) = 8

Solving
21y + 6y2 = 8

Solving for variable 'y'.

Reorder the terms:
-8 + 21y + 6y2 = 8 + -8

Combine like terms: 8 + -8 = 0
-8 + 21y + 6y2 = 0

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

Divide each side by '6'.
-1.333333333 + 3.5y + y2 = 0

Move the constant term to the right:

Add '1.333333333' to each side of the equation.
-1.333333333 + 3.5y + 1.333333333 + y2 = 0 + 1.333333333

Reorder the terms:
-1.333333333 + 1.333333333 + 3.5y + y2 = 0 + 1.333333333

Combine like terms: -1.333333333 + 1.333333333 = 0.000000000
0.000000000 + 3.5y + y2 = 0 + 1.333333333
3.5y + y2 = 0 + 1.333333333

Combine like terms: 0 + 1.333333333 = 1.333333333
3.5y + y2 = 1.333333333

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 = 1.333333333 + 3.0625

Reorder the terms:
3.0625 + 3.5y + y2 = 1.333333333 + 3.0625

Combine like terms: 1.333333333 + 3.0625 = 4.395833333
3.0625 + 3.5y + y2 = 4.395833333

Factor a perfect square on the left side:
(y + 1.75)(y + 1.75) = 4.395833333

Calculate the square root of the right side: 2.096624271

Break this problem into two subproblems by setting 
(y + 1.75) equal to 2.096624271 and -2.096624271.


Subproblem 1
y + 1.75 = 2.096624271

Simplifying
y + 1.75 = 2.096624271

Reorder the terms:
1.75 + y = 2.096624271

Solving
1.75 + y = 2.096624271

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 = 2.096624271 + -1.75

Combine like terms: 1.75 + -1.75 = 0.00
0.00 + y = 2.096624271 + -1.75
y = 2.096624271 + -1.75

Combine like terms: 2.096624271 + -1.75 = 0.346624271
y = 0.346624271

Simplifying
y = 0.346624271


Subproblem 2
y + 1.75 = -2.096624271

Simplifying
y + 1.75 = -2.096624271

Reorder the terms:
1.75 + y = -2.096624271

Solving
1.75 + y = -2.096624271

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 = -2.096624271 + -1.75

Combine like terms: 1.75 + -1.75 = 0.00
0.00 + y = -2.096624271 + -1.75
y = -2.096624271 + -1.75

Combine like terms: -2.096624271 + -1.75 = -3.846624271
y = -3.846624271

Simplifying
y = -3.846624271


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {0.346624271, -3.846624271}

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