0=3x^2-9x-27

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Solution for 0=3x^2-9x-27 equation: 


Simplifying
0 = 3x2 + -9x + -27

Reorder the terms:
0 = -27 + -9x + 3x2

Solving
0 = -27 + -9x + 3x2

Solving for variable 'x'.

Combine like terms: 0 + 27 = 27
27 + 9x + -3x2 = -27 + -9x + 3x2 + 27 + 9x + -3x2

Reorder the terms:
27 + 9x + -3x2 = -27 + 27 + -9x + 9x + 3x2 + -3x2

Combine like terms: -27 + 27 = 0
27 + 9x + -3x2 = 0 + -9x + 9x + 3x2 + -3x2
27 + 9x + -3x2 = -9x + 9x + 3x2 + -3x2

Combine like terms: -9x + 9x = 0
27 + 9x + -3x2 = 0 + 3x2 + -3x2
27 + 9x + -3x2 = 3x2 + -3x2

Combine like terms: 3x2 + -3x2 = 0
27 + 9x + -3x2 = 0

Factor out the Greatest Common Factor (GCF), '3'.
3(9 + 3x + -1x2) = 0

Ignore the factor 3.

Subproblem 1
Set the factor '(9 + 3x + -1x2)' equal to zero and attempt to solve:

Simplifying
9 + 3x + -1x2 = 0

Solving
9 + 3x + -1x2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-9 + -3x + x2 = 0

Move the constant term to the right:

Add '9' to each side of the equation.
-9 + -3x + 9 + x2 = 0 + 9

Reorder the terms:
-9 + 9 + -3x + x2 = 0 + 9

Combine like terms: -9 + 9 = 0
0 + -3x + x2 = 0 + 9
-3x + x2 = 0 + 9

Combine like terms: 0 + 9 = 9
-3x + x2 = 9

The x term is -3x.  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.
-3x + 2.25 + x2 = 9 + 2.25

Reorder the terms:
2.25 + -3x + x2 = 9 + 2.25

Combine like terms: 9 + 2.25 = 11.25
2.25 + -3x + x2 = 11.25

Factor a perfect square on the left side:
(x + -1.5)(x + -1.5) = 11.25

Calculate the square root of the right side: 3.354101966

Break this problem into two subproblems by setting 
(x + -1.5) equal to 3.354101966 and -3.354101966.


Subproblem 1
x + -1.5 = 3.354101966

Simplifying
x + -1.5 = 3.354101966

Reorder the terms:
-1.5 + x = 3.354101966

Solving
-1.5 + x = 3.354101966

Solving for variable 'x'.

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

Add '1.5' to each side of the equation.
-1.5 + 1.5 + x = 3.354101966 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + x = 3.354101966 + 1.5
x = 3.354101966 + 1.5

Combine like terms: 3.354101966 + 1.5 = 4.854101966
x = 4.854101966

Simplifying
x = 4.854101966


Subproblem 2
x + -1.5 = -3.354101966

Simplifying
x + -1.5 = -3.354101966

Reorder the terms:
-1.5 + x = -3.354101966

Solving
-1.5 + x = -3.354101966

Solving for variable 'x'.

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

Add '1.5' to each side of the equation.
-1.5 + 1.5 + x = -3.354101966 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + x = -3.354101966 + 1.5
x = -3.354101966 + 1.5

Combine like terms: -3.354101966 + 1.5 = -1.854101966
x = -1.854101966

Simplifying
x = -1.854101966


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {4.854101966, -1.854101966}
Solution
x = {4.854101966, -1.854101966}

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