3x^2-27x-54=0

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Solution for 3x^2-27x-54=0 equation: 


Simplifying
3x2 + -27x + -54 = 0

Reorder the terms:
-54 + -27x + 3x2 = 0

Solving
-54 + -27x + 3x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '3'.
3(-18 + -9x + x2) = 0

Ignore the factor 3.

Subproblem 1
Set the factor '(-18 + -9x + x2)' equal to zero and attempt to solve:

Simplifying
-18 + -9x + x2 = 0

Solving
-18 + -9x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '18' to each side of the equation.
-18 + -9x + 18 + x2 = 0 + 18

Reorder the terms:
-18 + 18 + -9x + x2 = 0 + 18

Combine like terms: -18 + 18 = 0
0 + -9x + x2 = 0 + 18
-9x + x2 = 0 + 18

Combine like terms: 0 + 18 = 18
-9x + x2 = 18

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 = 18 + 20.25

Reorder the terms:
20.25 + -9x + x2 = 18 + 20.25

Combine like terms: 18 + 20.25 = 38.25
20.25 + -9x + x2 = 38.25

Factor a perfect square on the left side:
(x + -4.5)(x + -4.5) = 38.25

Calculate the square root of the right side: 6.184658438

Break this problem into two subproblems by setting 
(x + -4.5) equal to 6.184658438 and -6.184658438.


Subproblem 1
x + -4.5 = 6.184658438

Simplifying
x + -4.5 = 6.184658438

Reorder the terms:
-4.5 + x = 6.184658438

Solving
-4.5 + x = 6.184658438

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 = 6.184658438 + 4.5

Combine like terms: -4.5 + 4.5 = 0.0
0.0 + x = 6.184658438 + 4.5
x = 6.184658438 + 4.5

Combine like terms: 6.184658438 + 4.5 = 10.684658438
x = 10.684658438

Simplifying
x = 10.684658438


Subproblem 2
x + -4.5 = -6.184658438

Simplifying
x + -4.5 = -6.184658438

Reorder the terms:
-4.5 + x = -6.184658438

Solving
-4.5 + x = -6.184658438

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 = -6.184658438 + 4.5

Combine like terms: -4.5 + 4.5 = 0.0
0.0 + x = -6.184658438 + 4.5
x = -6.184658438 + 4.5

Combine like terms: -6.184658438 + 4.5 = -1.684658438
x = -1.684658438

Simplifying
x = -1.684658438


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {10.684658438, -1.684658438}
Solution
x = {10.684658438, -1.684658438}

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