3x^2-27x-135=0

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Solution for 3x^2-27x-135=0 equation: 


Simplifying
3x2 + -27x + -135 = 0

Reorder the terms:
-135 + -27x + 3x2 = 0

Solving
-135 + -27x + 3x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '3'.
3(-45 + -9x + x2) = 0

Ignore the factor 3.

Subproblem 1
Set the factor '(-45 + -9x + x2)' equal to zero and attempt to solve:

Simplifying
-45 + -9x + x2 = 0

Solving
-45 + -9x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '45' to each side of the equation.
-45 + -9x + 45 + x2 = 0 + 45

Reorder the terms:
-45 + 45 + -9x + x2 = 0 + 45

Combine like terms: -45 + 45 = 0
0 + -9x + x2 = 0 + 45
-9x + x2 = 0 + 45

Combine like terms: 0 + 45 = 45
-9x + x2 = 45

The x term is -9x.  Take half its coefficient (-4.5).
Square it (20.25) and add it to both sides.

Add '20.25' to each side of the equation.
-9x + 20.25 + x2 = 45 + 20.25

Reorder the terms:
20.25 + -9x + x2 = 45 + 20.25

Combine like terms: 45 + 20.25 = 65.25
20.25 + -9x + x2 = 65.25

Factor a perfect square on the left side:
(x + -4.5)(x + -4.5) = 65.25

Calculate the square root of the right side: 8.077747211

Break this problem into two subproblems by setting 
(x + -4.5) equal to 8.077747211 and -8.077747211.


Subproblem 1
x + -4.5 = 8.077747211

Simplifying
x + -4.5 = 8.077747211

Reorder the terms:
-4.5 + x = 8.077747211

Solving
-4.5 + x = 8.077747211

Solving for variable 'x'.

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

Add '4.5' to each side of the equation.
-4.5 + 4.5 + x = 8.077747211 + 4.5

Combine like terms: -4.5 + 4.5 = 0.0
0.0 + x = 8.077747211 + 4.5
x = 8.077747211 + 4.5

Combine like terms: 8.077747211 + 4.5 = 12.577747211
x = 12.577747211

Simplifying
x = 12.577747211


Subproblem 2
x + -4.5 = -8.077747211

Simplifying
x + -4.5 = -8.077747211

Reorder the terms:
-4.5 + x = -8.077747211

Solving
-4.5 + x = -8.077747211

Solving for variable 'x'.

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

Add '4.5' to each side of the equation.
-4.5 + 4.5 + x = -8.077747211 + 4.5

Combine like terms: -4.5 + 4.5 = 0.0
0.0 + x = -8.077747211 + 4.5
x = -8.077747211 + 4.5

Combine like terms: -8.077747211 + 4.5 = -3.577747211
x = -3.577747211

Simplifying
x = -3.577747211


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {12.577747211, -3.577747211}
Solution
x = {12.577747211, -3.577747211}

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