3y^2+23y+16=0

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Solution for 3y^2+23y+16=0 equation: 


Simplifying
3y2 + 23y + 16 = 0

Reorder the terms:
16 + 23y + 3y2 = 0

Solving
16 + 23y + 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'.
5.333333333 + 7.666666667y + y2 = 0

Move the constant term to the right:

Add '-5.333333333' to each side of the equation.
5.333333333 + 7.666666667y + -5.333333333 + y2 = 0 + -5.333333333

Reorder the terms:
5.333333333 + -5.333333333 + 7.666666667y + y2 = 0 + -5.333333333

Combine like terms: 5.333333333 + -5.333333333 = 0.000000000
0.000000000 + 7.666666667y + y2 = 0 + -5.333333333
7.666666667y + y2 = 0 + -5.333333333

Combine like terms: 0 + -5.333333333 = -5.333333333
7.666666667y + y2 = -5.333333333

The y term is 7.666666667y.  Take half its coefficient (3.833333334).
Square it (14.69444445) and add it to both sides.

Add '14.69444445' to each side of the equation.
7.666666667y + 14.69444445 + y2 = -5.333333333 + 14.69444445

Reorder the terms:
14.69444445 + 7.666666667y + y2 = -5.333333333 + 14.69444445

Combine like terms: -5.333333333 + 14.69444445 = 9.361111117
14.69444445 + 7.666666667y + y2 = 9.361111117

Factor a perfect square on the left side:
(y + 3.833333334)(y + 3.833333334) = 9.361111117

Calculate the square root of the right side: 3.059593293

Break this problem into two subproblems by setting 
(y + 3.833333334) equal to 3.059593293 and -3.059593293.


Subproblem 1
y + 3.833333334 = 3.059593293

Simplifying
y + 3.833333334 = 3.059593293

Reorder the terms:
3.833333334 + y = 3.059593293

Solving
3.833333334 + y = 3.059593293

Solving for variable 'y'.

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

Add '-3.833333334' to each side of the equation.
3.833333334 + -3.833333334 + y = 3.059593293 + -3.833333334

Combine like terms: 3.833333334 + -3.833333334 = 0.000000000
0.000000000 + y = 3.059593293 + -3.833333334
y = 3.059593293 + -3.833333334

Combine like terms: 3.059593293 + -3.833333334 = -0.773740041
y = -0.773740041

Simplifying
y = -0.773740041


Subproblem 2
y + 3.833333334 = -3.059593293

Simplifying
y + 3.833333334 = -3.059593293

Reorder the terms:
3.833333334 + y = -3.059593293

Solving
3.833333334 + y = -3.059593293

Solving for variable 'y'.

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

Add '-3.833333334' to each side of the equation.
3.833333334 + -3.833333334 + y = -3.059593293 + -3.833333334

Combine like terms: 3.833333334 + -3.833333334 = 0.000000000
0.000000000 + y = -3.059593293 + -3.833333334
y = -3.059593293 + -3.833333334

Combine like terms: -3.059593293 + -3.833333334 = -6.892926627
y = -6.892926627

Simplifying
y = -6.892926627


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {-0.773740041, -6.892926627}

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