4(3y^2+22y+4)=1

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Solution for 4(3y^2+22y+4)=1 equation: 


Simplifying
4(3y2 + 22y + 4) = 1

Reorder the terms:
4(4 + 22y + 3y2) = 1
(4 * 4 + 22y * 4 + 3y2 * 4) = 1
(16 + 88y + 12y2) = 1

Solving
16 + 88y + 12y2 = 1

Solving for variable 'y'.

Reorder the terms:
16 + -1 + 88y + 12y2 = 1 + -1

Combine like terms: 16 + -1 = 15
15 + 88y + 12y2 = 1 + -1

Combine like terms: 1 + -1 = 0
15 + 88y + 12y2 = 0

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

Divide each side by '12'.
1.25 + 7.333333333y + y2 = 0

Move the constant term to the right:

Add '-1.25' to each side of the equation.
1.25 + 7.333333333y + -1.25 + y2 = 0 + -1.25

Reorder the terms:
1.25 + -1.25 + 7.333333333y + y2 = 0 + -1.25

Combine like terms: 1.25 + -1.25 = 0.00
0.00 + 7.333333333y + y2 = 0 + -1.25
7.333333333y + y2 = 0 + -1.25

Combine like terms: 0 + -1.25 = -1.25
7.333333333y + y2 = -1.25

The y term is 7.333333333y.  Take half its coefficient (3.666666667).
Square it (13.44444445) and add it to both sides.

Add '13.44444445' to each side of the equation.
7.333333333y + 13.44444445 + y2 = -1.25 + 13.44444445

Reorder the terms:
13.44444445 + 7.333333333y + y2 = -1.25 + 13.44444445

Combine like terms: -1.25 + 13.44444445 = 12.19444445
13.44444445 + 7.333333333y + y2 = 12.19444445

Factor a perfect square on the left side:
(y + 3.666666667)(y + 3.666666667) = 12.19444445

Calculate the square root of the right side: 3.492054474

Break this problem into two subproblems by setting 
(y + 3.666666667) equal to 3.492054474 and -3.492054474.


Subproblem 1
y + 3.666666667 = 3.492054474

Simplifying
y + 3.666666667 = 3.492054474

Reorder the terms:
3.666666667 + y = 3.492054474

Solving
3.666666667 + y = 3.492054474

Solving for variable 'y'.

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

Add '-3.666666667' to each side of the equation.
3.666666667 + -3.666666667 + y = 3.492054474 + -3.666666667

Combine like terms: 3.666666667 + -3.666666667 = 0.000000000
0.000000000 + y = 3.492054474 + -3.666666667
y = 3.492054474 + -3.666666667

Combine like terms: 3.492054474 + -3.666666667 = -0.174612193
y = -0.174612193

Simplifying
y = -0.174612193


Subproblem 2
y + 3.666666667 = -3.492054474

Simplifying
y + 3.666666667 = -3.492054474

Reorder the terms:
3.666666667 + y = -3.492054474

Solving
3.666666667 + y = -3.492054474

Solving for variable 'y'.

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

Add '-3.666666667' to each side of the equation.
3.666666667 + -3.666666667 + y = -3.492054474 + -3.666666667

Combine like terms: 3.666666667 + -3.666666667 = 0.000000000
0.000000000 + y = -3.492054474 + -3.666666667
y = -3.492054474 + -3.666666667

Combine like terms: -3.492054474 + -3.666666667 = -7.158721141
y = -7.158721141

Simplifying
y = -7.158721141


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {-0.174612193, -7.158721141}

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