-6y^2-16y+45=0

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Solution for -6y^2-16y+45=0 equation: 


Simplifying
-6y2 + -16y + 45 = 0

Reorder the terms:
45 + -16y + -6y2 = 0

Solving
45 + -16y + -6y2 = 0

Solving for variable 'y'.

Begin completing the square.  Divide all terms by
-6 the coefficient of the squared term: 

Divide each side by '-6'.
-7.5 + 2.666666667y + y2 = 0

Move the constant term to the right:

Add '7.5' to each side of the equation.
-7.5 + 2.666666667y + 7.5 + y2 = 0 + 7.5

Reorder the terms:
-7.5 + 7.5 + 2.666666667y + y2 = 0 + 7.5

Combine like terms: -7.5 + 7.5 = 0.0
0.0 + 2.666666667y + y2 = 0 + 7.5
2.666666667y + y2 = 0 + 7.5

Combine like terms: 0 + 7.5 = 7.5
2.666666667y + y2 = 7.5

The y term is 2.666666667y.  Take half its coefficient (1.333333334).
Square it (1.777777780) and add it to both sides.

Add '1.777777780' to each side of the equation.
2.666666667y + 1.777777780 + y2 = 7.5 + 1.777777780

Reorder the terms:
1.777777780 + 2.666666667y + y2 = 7.5 + 1.777777780

Combine like terms: 7.5 + 1.777777780 = 9.27777778
1.777777780 + 2.666666667y + y2 = 9.27777778

Factor a perfect square on the left side:
(y + 1.333333334)(y + 1.333333334) = 9.27777778

Calculate the square root of the right side: 3.045944481

Break this problem into two subproblems by setting 
(y + 1.333333334) equal to 3.045944481 and -3.045944481.


Subproblem 1
y + 1.333333334 = 3.045944481

Simplifying
y + 1.333333334 = 3.045944481

Reorder the terms:
1.333333334 + y = 3.045944481

Solving
1.333333334 + y = 3.045944481

Solving for variable 'y'.

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

Add '-1.333333334' to each side of the equation.
1.333333334 + -1.333333334 + y = 3.045944481 + -1.333333334

Combine like terms: 1.333333334 + -1.333333334 = 0.000000000
0.000000000 + y = 3.045944481 + -1.333333334
y = 3.045944481 + -1.333333334

Combine like terms: 3.045944481 + -1.333333334 = 1.712611147
y = 1.712611147

Simplifying
y = 1.712611147


Subproblem 2
y + 1.333333334 = -3.045944481

Simplifying
y + 1.333333334 = -3.045944481

Reorder the terms:
1.333333334 + y = -3.045944481

Solving
1.333333334 + y = -3.045944481

Solving for variable 'y'.

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

Add '-1.333333334' to each side of the equation.
1.333333334 + -1.333333334 + y = -3.045944481 + -1.333333334

Combine like terms: 1.333333334 + -1.333333334 = 0.000000000
0.000000000 + y = -3.045944481 + -1.333333334
y = -3.045944481 + -1.333333334

Combine like terms: -3.045944481 + -1.333333334 = -4.379277815
y = -4.379277815

Simplifying
y = -4.379277815


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {1.712611147, -4.379277815}

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