3y^2-14y=16

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Solution for 3y^2-14y=16 equation: 


Simplifying
3y2 + -14y = 16

Reorder the terms:
-14y + 3y2 = 16

Solving
-14y + 3y2 = 16

Solving for variable 'y'.

Reorder the terms:
-16 + -14y + 3y2 = 16 + -16

Combine like terms: 16 + -16 = 0
-16 + -14y + 3y2 = 0

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

Divide each side by '3'.
-5.333333333 + -4.666666667y + y2 = 0

Move the constant term to the right:

Add '5.333333333' to each side of the equation.
-5.333333333 + -4.666666667y + 5.333333333 + y2 = 0 + 5.333333333

Reorder the terms:
-5.333333333 + 5.333333333 + -4.666666667y + y2 = 0 + 5.333333333

Combine like terms: -5.333333333 + 5.333333333 = 0.000000000
0.000000000 + -4.666666667y + y2 = 0 + 5.333333333
-4.666666667y + y2 = 0 + 5.333333333

Combine like terms: 0 + 5.333333333 = 5.333333333
-4.666666667y + y2 = 5.333333333

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 = 5.333333333 + 5.444444448

Reorder the terms:
5.444444448 + -4.666666667y + y2 = 5.333333333 + 5.444444448

Combine like terms: 5.333333333 + 5.444444448 = 10.777777781
5.444444448 + -4.666666667y + y2 = 10.777777781

Factor a perfect square on the left side:
(y + -2.333333334)(y + -2.333333334) = 10.777777781

Calculate the square root of the right side: 3.282952601

Break this problem into two subproblems by setting 
(y + -2.333333334) equal to 3.282952601 and -3.282952601.


Subproblem 1
y + -2.333333334 = 3.282952601

Simplifying
y + -2.333333334 = 3.282952601

Reorder the terms:
-2.333333334 + y = 3.282952601

Solving
-2.333333334 + y = 3.282952601

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 = 3.282952601 + 2.333333334

Combine like terms: -2.333333334 + 2.333333334 = 0.000000000
0.000000000 + y = 3.282952601 + 2.333333334
y = 3.282952601 + 2.333333334

Combine like terms: 3.282952601 + 2.333333334 = 5.616285935
y = 5.616285935

Simplifying
y = 5.616285935


Subproblem 2
y + -2.333333334 = -3.282952601

Simplifying
y + -2.333333334 = -3.282952601

Reorder the terms:
-2.333333334 + y = -3.282952601

Solving
-2.333333334 + y = -3.282952601

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 = -3.282952601 + 2.333333334

Combine like terms: -2.333333334 + 2.333333334 = 0.000000000
0.000000000 + y = -3.282952601 + 2.333333334
y = -3.282952601 + 2.333333334

Combine like terms: -3.282952601 + 2.333333334 = -0.949619267
y = -0.949619267

Simplifying
y = -0.949619267


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {5.616285935, -0.949619267}

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