2y^2-6y-54=0

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Solution for 2y^2-6y-54=0 equation: 


Simplifying
2y2 + -6y + -54 = 0

Reorder the terms:
-54 + -6y + 2y2 = 0

Solving
-54 + -6y + 2y2 = 0

Solving for variable 'y'.

Factor out the Greatest Common Factor (GCF), '2'.
2(-27 + -3y + y2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-27 + -3y + y2)' equal to zero and attempt to solve:

Simplifying
-27 + -3y + y2 = 0

Solving
-27 + -3y + y2 = 0

Begin completing the square.

Move the constant term to the right:

Add '27' to each side of the equation.
-27 + -3y + 27 + y2 = 0 + 27

Reorder the terms:
-27 + 27 + -3y + y2 = 0 + 27

Combine like terms: -27 + 27 = 0
0 + -3y + y2 = 0 + 27
-3y + y2 = 0 + 27

Combine like terms: 0 + 27 = 27
-3y + y2 = 27

The y term is -3y.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3y + 2.25 + y2 = 27 + 2.25

Reorder the terms:
2.25 + -3y + y2 = 27 + 2.25

Combine like terms: 27 + 2.25 = 29.25
2.25 + -3y + y2 = 29.25

Factor a perfect square on the left side:
(y + -1.5)(y + -1.5) = 29.25

Calculate the square root of the right side: 5.408326913

Break this problem into two subproblems by setting 
(y + -1.5) equal to 5.408326913 and -5.408326913.


Subproblem 1
y + -1.5 = 5.408326913

Simplifying
y + -1.5 = 5.408326913

Reorder the terms:
-1.5 + y = 5.408326913

Solving
-1.5 + y = 5.408326913

Solving for variable 'y'.

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

Add '1.5' to each side of the equation.
-1.5 + 1.5 + y = 5.408326913 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + y = 5.408326913 + 1.5
y = 5.408326913 + 1.5

Combine like terms: 5.408326913 + 1.5 = 6.908326913
y = 6.908326913

Simplifying
y = 6.908326913


Subproblem 2
y + -1.5 = -5.408326913

Simplifying
y + -1.5 = -5.408326913

Reorder the terms:
-1.5 + y = -5.408326913

Solving
-1.5 + y = -5.408326913

Solving for variable 'y'.

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

Add '1.5' to each side of the equation.
-1.5 + 1.5 + y = -5.408326913 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + y = -5.408326913 + 1.5
y = -5.408326913 + 1.5

Combine like terms: -5.408326913 + 1.5 = -3.908326913
y = -3.908326913

Simplifying
y = -3.908326913


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {6.908326913, -3.908326913}
Solution
y = {6.908326913, -3.908326913}

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