6y^2-27y-45=0

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Solution for 6y^2-27y-45=0 equation: 


Simplifying
6y2 + -27y + -45 = 0

Reorder the terms:
-45 + -27y + 6y2 = 0

Solving
-45 + -27y + 6y2 = 0

Solving for variable 'y'.

Factor out the Greatest Common Factor (GCF), '3'.
3(-15 + -9y + 2y2) = 0

Ignore the factor 3.

Subproblem 1
Set the factor '(-15 + -9y + 2y2)' equal to zero and attempt to solve:

Simplifying
-15 + -9y + 2y2 = 0

Solving
-15 + -9y + 2y2 = 0

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

Divide each side by '2'.
-7.5 + -4.5y + y2 = 0

Move the constant term to the right:

Add '7.5' to each side of the equation.
-7.5 + -4.5y + 7.5 + y2 = 0 + 7.5

Reorder the terms:
-7.5 + 7.5 + -4.5y + y2 = 0 + 7.5

Combine like terms: -7.5 + 7.5 = 0.0
0.0 + -4.5y + y2 = 0 + 7.5
-4.5y + y2 = 0 + 7.5

Combine like terms: 0 + 7.5 = 7.5
-4.5y + y2 = 7.5

The y term is -4.5y.  Take half its coefficient (-2.25).
Square it (5.0625) and add it to both sides.

Add '5.0625' to each side of the equation.
-4.5y + 5.0625 + y2 = 7.5 + 5.0625

Reorder the terms:
5.0625 + -4.5y + y2 = 7.5 + 5.0625

Combine like terms: 7.5 + 5.0625 = 12.5625
5.0625 + -4.5y + y2 = 12.5625

Factor a perfect square on the left side:
(y + -2.25)(y + -2.25) = 12.5625

Calculate the square root of the right side: 3.54436172

Break this problem into two subproblems by setting 
(y + -2.25) equal to 3.54436172 and -3.54436172.


Subproblem 1
y + -2.25 = 3.54436172

Simplifying
y + -2.25 = 3.54436172

Reorder the terms:
-2.25 + y = 3.54436172

Solving
-2.25 + y = 3.54436172

Solving for variable 'y'.

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

Add '2.25' to each side of the equation.
-2.25 + 2.25 + y = 3.54436172 + 2.25

Combine like terms: -2.25 + 2.25 = 0.00
0.00 + y = 3.54436172 + 2.25
y = 3.54436172 + 2.25

Combine like terms: 3.54436172 + 2.25 = 5.79436172
y = 5.79436172

Simplifying
y = 5.79436172


Subproblem 2
y + -2.25 = -3.54436172

Simplifying
y + -2.25 = -3.54436172

Reorder the terms:
-2.25 + y = -3.54436172

Solving
-2.25 + y = -3.54436172

Solving for variable 'y'.

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

Add '2.25' to each side of the equation.
-2.25 + 2.25 + y = -3.54436172 + 2.25

Combine like terms: -2.25 + 2.25 = 0.00
0.00 + y = -3.54436172 + 2.25
y = -3.54436172 + 2.25

Combine like terms: -3.54436172 + 2.25 = -1.29436172
y = -1.29436172

Simplifying
y = -1.29436172


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {5.79436172, -1.29436172}
Solution
y = {5.79436172, -1.29436172}

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