3y^2+14y+4=0

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Solution for 3y^2+14y+4=0 equation: 


Simplifying
3y2 + 14y + 4 = 0

Reorder the terms:
4 + 14y + 3y2 = 0

Solving
4 + 14y + 3y2 = 0

Solving for variable 'y'.

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

Divide each side by '3'.
1.333333333 + 4.666666667y + y2 = 0

Move the constant term to the right:

Add '-1.333333333' to each side of the equation.
1.333333333 + 4.666666667y + -1.333333333 + y2 = 0 + -1.333333333

Reorder the terms:
1.333333333 + -1.333333333 + 4.666666667y + y2 = 0 + -1.333333333

Combine like terms: 1.333333333 + -1.333333333 = 0.000000000
0.000000000 + 4.666666667y + y2 = 0 + -1.333333333
4.666666667y + y2 = 0 + -1.333333333

Combine like terms: 0 + -1.333333333 = -1.333333333
4.666666667y + y2 = -1.333333333

The y term is 4.666666667y.  Take half its coefficient (2.333333334).
Square it (5.444444448) and add it to both sides.

Add '5.444444448' to each side of the equation.
4.666666667y + 5.444444448 + y2 = -1.333333333 + 5.444444448

Reorder the terms:
5.444444448 + 4.666666667y + y2 = -1.333333333 + 5.444444448

Combine like terms: -1.333333333 + 5.444444448 = 4.111111115
5.444444448 + 4.666666667y + y2 = 4.111111115

Factor a perfect square on the left side:
(y + 2.333333334)(y + 2.333333334) = 4.111111115

Calculate the square root of the right side: 2.027587511

Break this problem into two subproblems by setting 
(y + 2.333333334) equal to 2.027587511 and -2.027587511.


Subproblem 1
y + 2.333333334 = 2.027587511

Simplifying
y + 2.333333334 = 2.027587511

Reorder the terms:
2.333333334 + y = 2.027587511

Solving
2.333333334 + y = 2.027587511

Solving for variable 'y'.

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

Add '-2.333333334' to each side of the equation.
2.333333334 + -2.333333334 + y = 2.027587511 + -2.333333334

Combine like terms: 2.333333334 + -2.333333334 = 0.000000000
0.000000000 + y = 2.027587511 + -2.333333334
y = 2.027587511 + -2.333333334

Combine like terms: 2.027587511 + -2.333333334 = -0.305745823
y = -0.305745823

Simplifying
y = -0.305745823


Subproblem 2
y + 2.333333334 = -2.027587511

Simplifying
y + 2.333333334 = -2.027587511

Reorder the terms:
2.333333334 + y = -2.027587511

Solving
2.333333334 + y = -2.027587511

Solving for variable 'y'.

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

Add '-2.333333334' to each side of the equation.
2.333333334 + -2.333333334 + y = -2.027587511 + -2.333333334

Combine like terms: 2.333333334 + -2.333333334 = 0.000000000
0.000000000 + y = -2.027587511 + -2.333333334
y = -2.027587511 + -2.333333334

Combine like terms: -2.027587511 + -2.333333334 = -4.360920845
y = -4.360920845

Simplifying
y = -4.360920845


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {-0.305745823, -4.360920845}

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