3y(2y+1)+2y(y+6)=25

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Solution for 3y(2y+1)+2y(y+6)=25 equation: 


Simplifying
3y(2y + 1) + 2y(y + 6) = 25

Reorder the terms:
3y(1 + 2y) + 2y(y + 6) = 25
(1 * 3y + 2y * 3y) + 2y(y + 6) = 25
(3y + 6y2) + 2y(y + 6) = 25

Reorder the terms:
3y + 6y2 + 2y(6 + y) = 25
3y + 6y2 + (6 * 2y + y * 2y) = 25
3y + 6y2 + (12y + 2y2) = 25

Reorder the terms:
3y + 12y + 6y2 + 2y2 = 25

Combine like terms: 3y + 12y = 15y
15y + 6y2 + 2y2 = 25

Combine like terms: 6y2 + 2y2 = 8y2
15y + 8y2 = 25

Solving
15y + 8y2 = 25

Solving for variable 'y'.

Reorder the terms:
-25 + 15y + 8y2 = 25 + -25

Combine like terms: 25 + -25 = 0
-25 + 15y + 8y2 = 0

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

Divide each side by '8'.
-3.125 + 1.875y + y2 = 0

Move the constant term to the right:

Add '3.125' to each side of the equation.
-3.125 + 1.875y + 3.125 + y2 = 0 + 3.125

Reorder the terms:
-3.125 + 3.125 + 1.875y + y2 = 0 + 3.125

Combine like terms: -3.125 + 3.125 = 0.000
0.000 + 1.875y + y2 = 0 + 3.125
1.875y + y2 = 0 + 3.125

Combine like terms: 0 + 3.125 = 3.125
1.875y + y2 = 3.125

The y term is 1.875y.  Take half its coefficient (0.9375).
Square it (0.87890625) and add it to both sides.

Add '0.87890625' to each side of the equation.
1.875y + 0.87890625 + y2 = 3.125 + 0.87890625

Reorder the terms:
0.87890625 + 1.875y + y2 = 3.125 + 0.87890625

Combine like terms: 3.125 + 0.87890625 = 4.00390625
0.87890625 + 1.875y + y2 = 4.00390625

Factor a perfect square on the left side:
(y + 0.9375)(y + 0.9375) = 4.00390625

Calculate the square root of the right side: 2.000976324

Break this problem into two subproblems by setting 
(y + 0.9375) equal to 2.000976324 and -2.000976324.


Subproblem 1
y + 0.9375 = 2.000976324

Simplifying
y + 0.9375 = 2.000976324

Reorder the terms:
0.9375 + y = 2.000976324

Solving
0.9375 + y = 2.000976324

Solving for variable 'y'.

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

Add '-0.9375' to each side of the equation.
0.9375 + -0.9375 + y = 2.000976324 + -0.9375

Combine like terms: 0.9375 + -0.9375 = 0.0000
0.0000 + y = 2.000976324 + -0.9375
y = 2.000976324 + -0.9375

Combine like terms: 2.000976324 + -0.9375 = 1.063476324
y = 1.063476324

Simplifying
y = 1.063476324


Subproblem 2
y + 0.9375 = -2.000976324

Simplifying
y + 0.9375 = -2.000976324

Reorder the terms:
0.9375 + y = -2.000976324

Solving
0.9375 + y = -2.000976324

Solving for variable 'y'.

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

Add '-0.9375' to each side of the equation.
0.9375 + -0.9375 + y = -2.000976324 + -0.9375

Combine like terms: 0.9375 + -0.9375 = 0.0000
0.0000 + y = -2.000976324 + -0.9375
y = -2.000976324 + -0.9375

Combine like terms: -2.000976324 + -0.9375 = -2.938476324
y = -2.938476324

Simplifying
y = -2.938476324


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {1.063476324, -2.938476324}

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