6x^2+3x-54=0

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Solution for 6x^2+3x-54=0 equation: 


Simplifying
6x2 + 3x + -54 = 0

Reorder the terms:
-54 + 3x + 6x2 = 0

Solving
-54 + 3x + 6x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '3'.
3(-18 + x + 2x2) = 0

Ignore the factor 3.

Subproblem 1
Set the factor '(-18 + x + 2x2)' equal to zero and attempt to solve:

Simplifying
-18 + x + 2x2 = 0

Solving
-18 + x + 2x2 = 0

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

Divide each side by '2'.
-9 + 0.5x + x2 = 0

Move the constant term to the right:

Add '9' to each side of the equation.
-9 + 0.5x + 9 + x2 = 0 + 9

Reorder the terms:
-9 + 9 + 0.5x + x2 = 0 + 9

Combine like terms: -9 + 9 = 0
0 + 0.5x + x2 = 0 + 9
0.5x + x2 = 0 + 9

Combine like terms: 0 + 9 = 9
0.5x + x2 = 9

The x term is x.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
0.5x + 0.25 + x2 = 9 + 0.25

Reorder the terms:
0.25 + 0.5x + x2 = 9 + 0.25

Combine like terms: 9 + 0.25 = 9.25
0.25 + 0.5x + x2 = 9.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 9.25

Calculate the square root of the right side: 3.041381265

Break this problem into two subproblems by setting 
(x + 0.5) equal to 3.041381265 and -3.041381265.


Subproblem 1
x + 0.5 = 3.041381265

Simplifying
x + 0.5 = 3.041381265

Reorder the terms:
0.5 + x = 3.041381265

Solving
0.5 + x = 3.041381265

Solving for variable 'x'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = 3.041381265 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 3.041381265 + -0.5
x = 3.041381265 + -0.5

Combine like terms: 3.041381265 + -0.5 = 2.541381265
x = 2.541381265

Simplifying
x = 2.541381265


Subproblem 2
x + 0.5 = -3.041381265

Simplifying
x + 0.5 = -3.041381265

Reorder the terms:
0.5 + x = -3.041381265

Solving
0.5 + x = -3.041381265

Solving for variable 'x'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = -3.041381265 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -3.041381265 + -0.5
x = -3.041381265 + -0.5

Combine like terms: -3.041381265 + -0.5 = -3.541381265
x = -3.541381265

Simplifying
x = -3.541381265


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.541381265, -3.541381265}
Solution
x = {2.541381265, -3.541381265}

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