18x^2-27x-54=0

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Solution for 18x^2-27x-54=0 equation: 


Simplifying
18x2 + -27x + -54 = 0

Reorder the terms:
-54 + -27x + 18x2 = 0

Solving
-54 + -27x + 18x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '9'.
9(-6 + -3x + 2x2) = 0

Ignore the factor 9.

Subproblem 1
Set the factor '(-6 + -3x + 2x2)' equal to zero and attempt to solve:

Simplifying
-6 + -3x + 2x2 = 0

Solving
-6 + -3x + 2x2 = 0

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

Divide each side by '2'.
-3 + -1.5x + x2 = 0

Move the constant term to the right:

Add '3' to each side of the equation.
-3 + -1.5x + 3 + x2 = 0 + 3

Reorder the terms:
-3 + 3 + -1.5x + x2 = 0 + 3

Combine like terms: -3 + 3 = 0
0 + -1.5x + x2 = 0 + 3
-1.5x + x2 = 0 + 3

Combine like terms: 0 + 3 = 3
-1.5x + x2 = 3

The x term is -1.5x.  Take half its coefficient (-0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
-1.5x + 0.5625 + x2 = 3 + 0.5625

Reorder the terms:
0.5625 + -1.5x + x2 = 3 + 0.5625

Combine like terms: 3 + 0.5625 = 3.5625
0.5625 + -1.5x + x2 = 3.5625

Factor a perfect square on the left side:
(x + -0.75)(x + -0.75) = 3.5625

Calculate the square root of the right side: 1.887458609

Break this problem into two subproblems by setting 
(x + -0.75) equal to 1.887458609 and -1.887458609.


Subproblem 1
x + -0.75 = 1.887458609

Simplifying
x + -0.75 = 1.887458609

Reorder the terms:
-0.75 + x = 1.887458609

Solving
-0.75 + x = 1.887458609

Solving for variable 'x'.

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

Add '0.75' to each side of the equation.
-0.75 + 0.75 + x = 1.887458609 + 0.75

Combine like terms: -0.75 + 0.75 = 0.00
0.00 + x = 1.887458609 + 0.75
x = 1.887458609 + 0.75

Combine like terms: 1.887458609 + 0.75 = 2.637458609
x = 2.637458609

Simplifying
x = 2.637458609


Subproblem 2
x + -0.75 = -1.887458609

Simplifying
x + -0.75 = -1.887458609

Reorder the terms:
-0.75 + x = -1.887458609

Solving
-0.75 + x = -1.887458609

Solving for variable 'x'.

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

Add '0.75' to each side of the equation.
-0.75 + 0.75 + x = -1.887458609 + 0.75

Combine like terms: -0.75 + 0.75 = 0.00
0.00 + x = -1.887458609 + 0.75
x = -1.887458609 + 0.75

Combine like terms: -1.887458609 + 0.75 = -1.137458609
x = -1.137458609

Simplifying
x = -1.137458609


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.637458609, -1.137458609}
Solution
x = {2.637458609, -1.137458609}

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